This is a multi-part message in MIME format. ---------------0203121354259 Content-Type: text/plain; name="02-118.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-118.keywords" continuous analogs of orthogonal polynomials on the circle; matrix canonical system of differential equations; density of the absolutely continuous component of the spectral measure; convergence of the adjoint polynomials; Szego condition (Szeg\"{o} condition); M. G. Krein; L. A. 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Fm(\))p Fl(f)11 b Fm(\()p Fl(t)p Fm(\))p Fl(dt:)1926 5484 y Fm(2)p eop %%Page: 3 3 3 2 bop 150 227 a Fm(The)34 b(co)s(e\016cien)m(t)f Fl(a)p Fm(\()p Fl(r)s Fm(\))f(of)g(the)h(equation)f(\(3.1\))g(is)h Fl(a)p Fm(\()p Fl(r)s Fm(\))27 b(=)h(\000)2387 242 y Fv(r)2424 227 y Fm(\(0)p Fl(;)17 b(r)s Fm(\).)296 347 y(If)25 b Fl(\034)439 311 y Fx(0)487 347 y Fm(is)f(the)h(densit)m(y)g (of)f(the)h(absolutely)f(con)m(tin)m(uous)h(comp)s(onen)m(t)f(of)g(the) h(sp)s(ectral)g(measure,)150 468 y(then)1497 569 y Fh(Z)1552 794 y Fy(R)1631 637 y Fi(j)17 b Fm(log)f Fl(\034)1871 601 y Fx(0)1894 637 y Fm(\()p Fl(\025)p Fm(\))p Fi(j)p 1631 682 425 4 v 1710 773 a Fm(1)22 b(+)g Fl(\025)1936 744 y Fw(2)2065 705 y Fl(d\025)27 b(<)h Fi(1)1145 b Fm(\(1.1\))150 958 y(if)31 b(and)i(only)f(if)1510 1075 y Fh(Z)1609 1101 y Fx(1)1565 1301 y Fw(0)1701 1211 y Fi(j)p Fl(p)p Fm(\()p Fl(r)m(;)17 b(\025)p Fm(\))p Fi(j)2024 1169 y Fw(2)2062 1211 y Fl(dr)30 b(<)e Fi(1)1158 b Fm(\(1.2\))150 1465 y(for)32 b(Im)p Fl(\025)27 b(>)h Fm(0.)k(In)h(the)g(cen)m(ter)h(of)e (our)g(discussion)h(is)f(the)h(existence)h(of)f(the)g(limit)1511 1672 y(\005\()p Fl(\025)p Fm(\))27 b(=)48 b(lim)1848 1731 y Fv(r)r Fx(!1)2039 1672 y Fl(p)2088 1630 y Fx(\003)2128 1672 y Fm(\()p Fl(r)m(;)17 b(\025)p Fm(\))p Fl(;)1176 b Fm(\(1.3\))150 1911 y(where)35 b(\005\()p Fl(\025)p Fm(\))e(is)g(analytic)f(for)h(Im)o Fl(\025)c(>)g Fm(0.)k(An)h(imp)s (ortan)m(t)e(relation,)f(whic)m(h)j(follo)m(ws)e(from)g(\(3.1\))150 2031 y(and)h(w)m(as)g(noted)g(b)m(y)h(M.)f(G.)f(Krein)g(in)g([K],)g(is) 1001 2289 y Fi(j)p Fl(p)1078 2248 y Fx(\003)1117 2289 y Fm(\()p Fl(r)m(;)17 b(\025)p Fm(\))p Fi(j)1363 2248 y Fw(2)1424 2289 y Fi(\000)23 b(j)p Fl(p)p Fm(\()p Fl(r)m(;)17 b(\025)p Fm(\))p Fi(j)1847 2248 y Fw(2)1913 2289 y Fm(=)27 b(2Im)p Fl(\025)2256 2153 y Fh(Z)2355 2180 y Fv(r)2310 2379 y Fw(0)2409 2289 y Fi(j)p Fl(p)p Fm(\()p Fl(s;)17 b(\025)p Fm(\))p Fi(j)2737 2248 y Fw(2)2775 2289 y Fl(ds:)650 b Fm(\(1.4\))150 2556 y(Th)m(us)34 b(w)m(e)g(m)m(ust)e(ha)m(v)m(e)i Fi(j)p Fm(\005\()p Fl(\025)p Fm(\))p Fi(j)1269 2520 y Fw(2)1336 2556 y Fm(=)27 b(2Im)p Fl(\025)1679 2476 y Fh(R)1744 2502 y Fx(1)1725 2591 y Fw(0)1836 2556 y Fi(j)p Fl(p)p Fm(\()p Fl(r)m(;)17 b(\025)p Fm(\))p Fi(j)2159 2520 y Fw(2)2197 2556 y Fl(dr)35 b Fm(if)d(the)h(in)m(tegral)e(con)m(v) m(erges.)296 2677 y(In)k(Section)f(6)g(w)m(e)h(sho)m(w)g(that)f(if)f (the)h(co)s(e\016cien)m(ts)i(are)e(square)h(in)m(tegrable,)e(then)i (the)f(limit)150 2797 y(\(1.3\))24 b(con)m(v)m(erges,)i(and)e(so)h (\005\()p Fl(\025)p Fm(\))f(is)f(uniquely)h(de\014ned.)i(The)f (existence)h(of)d(this)h(limit)d(implies)h(the)150 2917 y(con)m(v)m(ergence)30 b(of)e(the)g(in)m(tegrals)e(\(1.1\))h(and)h (\(1.2\),)f(but)h(the)h(con)m(v)m(erse)h(is)d(not)g(true)h(in)f (general.)h(In)150 3038 y(Section)c(4)g(w)m(e)i(pro)m(v)m(e)g(that)e (there)h(are)f(situations)g(when)h(\(1.1\))f(and)h(\(1.2\))f(hold,)f (but)i(\005\()p Fl(\025)p Fm(\))f(has)h(to)150 3158 y(b)s(e)32 b(de\014ned)g(as)g(a)f(limit)d(of)j(a)g(con)m(v)m(ergen)m(t)i (subsequence.)i(W)-8 b(e)32 b(sho)m(w)g(that)f(this)g(situation)f(is)h (not)150 3278 y(\\pathological",)c(but)k(can)g(o)s(ccur)g(ev)m(en)h(if) e(the)h(sp)s(ectral)g(measure)g Fl(\034)42 b Fm(is)30 b(absolutely)g(con)m(tin)m(uous)150 3399 y(with)g(p)s(ositiv)m(e)h(con) m(tin)m(uous)g(densit)m(y)h(\(Theorem)f(1\).)f(In)i(another)e(example)h (\(Theorem)g(2\),)f(this)150 3519 y(happ)s(ens)k(ev)m(en)g(though)799 3771 y Fi(j)p Fm(\005\()p Fl(\025)p Fm(\))p Fi(j)1061 3730 y Fw(2)1127 3771 y Fm(=)48 b(lim)1231 3831 y Fv(r)r Fx(!1)1422 3771 y Fi(j)p Fl(p)1499 3730 y Fx(\003)1538 3771 y Fm(\()p Fl(r)m(;)17 b(\025)p Fm(\))p Fi(j)1784 3730 y Fw(2)1851 3771 y Fm(=)28 b(2Im)o Fl(\025)2194 3636 y Fh(Z)2293 3662 y Fx(1)2249 3861 y Fw(0)2384 3771 y Fi(j)p Fl(p)p Fm(\()p Fl(r)m(;)17 b(\025)p Fm(\))p Fi(j)2707 3730 y Fw(2)2746 3771 y Fl(dr)30 b(<)d Fi(1)p Fl(;)150 4031 y Fm(and)34 b(the)f(co)s(e\016cien)m(ts)i(are)e(in)g Fl(L)1348 3995 y Fv(p)1421 4031 y Fm(for)g(an)m(y)h Fl(p)29 b(>)g Fm(2.)34 b(Moreo)m(v)m(er,)h(the)f(function)f(\005\()p Fl(\025)p Fm(\))g(can)g(not)h(b)s(e)150 4151 y(de\014ned)39 b(uniquely)-8 b(,)38 b(but)g(only)g(up)g(to)f(a)h(constan)m(t)h(factor) e(of)g(absolute)h(v)-5 b(alue)37 b(one)h(\(up)g(to)g(left)150 4272 y(m)m(ultiplication)d(b)m(y)40 b(a)f(unitary)g(matrix)f(in)h(the)g (case)i(of)e(Sakhno)m(vic)m(h's)i(theorem\).)e(Section)g(6)150 4392 y(con)m(tains)30 b(sev)m(eral)g(results)g(related)f(to)g(the)h (con)m(v)m(ergence)j(of)c(the)h(limit)c(\(1.3\))j(in)g(the)h(case)g(of) g(the)150 4512 y(Sakhno)m(vic)m(h's)k(di\013eren)m(tial)d(equations.) 296 4633 y(Note)44 b(that)g(results)g(of)f(Section)g(6)h(apply)f(to)g (Krein's)h(system,)h(since)f(it)e(is)h(a)h(particular)150 4753 y(case)34 b(of)e(Sakhno)m(vic)m(h's)j(system.)f(Tw)m(o)g(of)f(the) g(three)h(results)f(there)h(are)f(new)h(ev)m(en)h(for)d(Krein's)150 4874 y(system.)k(A)m(t)e(the)h(same)g(time)e(results)i(of)f(Section)g (4)g(are)h(stated)g(for)f(Krein's)h(system,)g(but)g(are)150 4994 y(applicable)c(for)h(Sakhno)m(vic)m(h's)i(system)g(as)e(w)m(ell.) 296 5114 y(The)27 b(fundamen)m(tal)d(pap)s(er)i([K])f(presen)m(ts)j(a)d (n)m(um)m(b)s(er)h(of)f(imp)s(ortan)m(t)f(results,)i(though)f(it)f(do)s (es)150 5235 y(not)32 b(con)m(tain)f(pro)s(ofs)g(due)i(to)e(the)h(t)m (yp)s(e)h(of)e(the)i(journal)d(it)h(w)m(as)i(published)e(in.)g(Later)h (pro)s(ofs)f(of)1926 5484 y(3)p eop %%Page: 4 4 4 3 bop 150 227 a Fm(Krein's)25 b(results)h(w)m(ere)h(giv)m(en)f(indep) s(enden)m(tly)g(b)m(y)h(the)f(author)f(in)g(1990)f(\([T1],)i(partly)f (published)150 347 y(in)30 b([T2]\))i(and)f(L.)g(A.)g(Sakhno)m(vic)m(h) h(in)f(1998)f(\([S2{S4]\).)h(The)h(main)d(sub)5 b(ject)33 b(of)e([T1,T2])g(w)m(as)h(to)150 468 y(pro)m(v)m(e)e(that)f(the)g(sp)s (ectral)f(measure)i Fl(\034)40 b Fm(is)28 b(absolutely)g(con)m(tin)m (uous)i(with)e(probabilit)m(y)f(one)i(if)f(the)150 588 y(co)s(e\016cien)m(t)33 b Fl(a)p Fm(\()p Fl(r)s Fm(\))f(is)h(a)f (random)g(function)g(satisfying)f(certain)h(conditions.)296 708 y(In)48 b([T1,T2])g(the)g(author)g(noted)g(and)f(corrected)i(an)f (inaccuracy)g(in)e(the)i(statemen)m(t)g(of)150 829 y(Krein's)36 b(theorem,)g(and)h(ga)m(v)m(e)g(a)f(pro)s(of)f(of)h(the)h(corrected)g (main)e(theorem)h(\(see)h(Section)f(3)g(for)150 949 y(more)d (details\).)f(Theorems)i(1)f(and)g(2)g(in)g(Section)g(4)g(pro)m(v)m(e,) h(in)f(particular,)f(that)h(a)g(part)g(of)g(the)150 1069 y(statemen)m(t)g(of)f(Krein's)h(theorem)f(in)g([K])g(needs)i(to)f(b)s (e)f(corrected.)296 1190 y(In)27 b([S1{S5])f(L.)g(A.)g(Sakhno)m(vic)m (h)h(de\014ned)h(and)e(studied)g(matrix)f(v)-5 b(alued)25 b(con)m(tin)m(uous)i(analogs)150 1310 y(of)41 b(orthogonal)g(p)s (olynomials)e(on)i(the)i(unit)e(circle,)g(and)h(pro)m(v)m(ed)i(matrix)c (generalizations)g(of)150 1431 y(Krein's)24 b(results.)h(Unfortunately) -8 b(,)24 b(these)i(w)m(orks)f(con)m(tain)f(the)h(same)f(kind)g(of)g (inaccuracy)g(as)h([K].)150 1551 y(In)32 b(Section)f(5)g(w)m(e)h (presen)m(t)h(the)f(corrected)g(statemen)m(t,)g(and)f(the)h(corrected)g (part)f(of)g(the)h(pro)s(of.)296 1671 y(W)-8 b(e)36 b(emphasize)g(that) g(the)g(inaccuracy)g(in)f(the)h(statemen)m(t)g(of)g(Krein's)f(and)h (Sakhno)m(vic)m(h's)150 1792 y(theorems)h(is)g(not)g(signi\014can)m(t,) f(and)i(do)s(es)f(not)g(undermine)g(kno)m(wn)h(implications)c(from)i (these)150 1912 y(imp)s(ortan)m(t)24 b(results.)i(F)-8 b(or)25 b(instance,)i(if)d(\(1.1\))i(and)g(\(1.2\))f(hold,)g(then)i (there)f(is)f(the)i(function)e(\005\()p Fl(\025)p Fm(\))150 2033 y(whic)m(h)33 b(is)f(analytic)f(and)i(has)g(no)g(zeros)g(for)f(Im) o Fl(\025)c(>)g Fm(0,)k(and)1565 2274 y Fl(\034)1618 2233 y Fx(0)1642 2274 y Fm(\()p Fl(\025)p Fm(\))c(=)2096 2206 y(1)p 1916 2251 409 4 v 1916 2342 a(2)p Fl(\031)t Fi(j)p Fm(\005\()p Fl(\025)p Fm(\))p Fi(j)2286 2313 y Fw(2)150 2532 y Fm(for)49 b(Leb)s(esgue)h(almost)e(all)f Fl(\025)56 b Fi(2)g Fk(R)k Fm(\(there)50 b(is)f(an)g(analogous)f (matrix)g(v)m(ersion)h(pro)m(v)m(ed)i(b)m(y)150 2653 y(L.)26 b(A.)f(Sakhno)m(vic)m(h)i(in)e([S4]\).)h(This)g(result)g (remains)e(unc)m(hanged)j(ev)m(en)h(if)c(the)i(limit)42 b(lim)3252 2712 y Fv(r)r Fx(!1)3444 2653 y Fl(p)3493 2616 y Fx(\003)3532 2653 y Fm(\()p Fl(r)m(;)17 b(\025)p Fm(\))150 2773 y(div)m(erges,)34 b(and)e(the)h(non)m(uniqueness)i(of)d (\005\()p Fl(\025)p Fm(\))h(men)m(tioned)f(ab)s(o)m(v)m(e)h(tak)m(es)h (place.)150 3030 y Fg(Ac)m(kno)m(wledgmen)m(ts.)150 3215 y Fm(The)39 b(author)e(is)g(deeply)h(grateful)f(to)g(I.)h(A.)g (Ibragimo)m(v)e(for)h(his)g(in)m(v)-5 b(aluable)36 b(guidance)i(during) 150 3335 y(an)29 b(early)g(part)g(of)g(this)f(w)m(ork,)j(whic)m(h)e(w)m (as)h(completed)f(in)f(St.-P)m(etersburg,)j(former)d(Leningrad,)150 3456 y(State)d(Univ)m(ersit)m(y)h(and)f(Steklo)m(v)h(Mathematical)d (Institute)i(\(POMI\).)h(The)g(author)f(also)f(thanks)150 3576 y(S.)h(A.)g(Deniso)m(v,)g(M.)h(I.)f(Gordin,)f(M.)h(L.)g(Lapidus)g (and)g(M.)g(Z.)g(Solom)m(y)m(ak)f(for)g(helpful)g(discussions)150 3696 y(during)32 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Fw(+1)1556 4831 y Fm(\()p Fl(z)t Fm(\))28 b(=)g(\010)1883 4790 y Fx(\003)1883 4856 y Fv(n)1930 4831 y Fm(\()p Fl(z)t Fm(\))23 b Fi(\000)f Fl(a)2228 4846 y Fv(n)2276 4831 y Fl(z)t Fm(\010)2395 4846 y Fv(n)2443 4831 y Fm(\()p Fl(z)t Fm(\))3549 4759 y(\(2.1\))150 5032 y(where)34 b Fi(f)p Fm(\010)552 4996 y Fx(\003)552 5057 y Fv(n)599 5032 y Fm(\()p Fl(z)t Fm(\))p Fi(g)774 4996 y Fx(1)774 5057 y Fv(n)p Fw(=0)944 5032 y Fm(is)e(an)h(auxiliary)d(sequence)35 b(of)e(p)s(olynomials)c(and)1545 5235 y(\010)1615 5250 y Fw(0)1655 5235 y Fm(\()p Fl(z)t Fm(\))g(=)e(\010)1982 5194 y Fx(\003)1982 5259 y Fw(0)2022 5235 y Fm(\()p Fl(z)t Fm(\))h(=)g(1)p Fl(:)1926 5484 y Fm(4)p eop %%Page: 5 5 5 4 bop 150 227 a Fm(The)33 b(p)s(olynomials)c(\010)962 191 y Fx(\003)962 251 y Fv(n)1010 227 y Fm(\()p Fl(z)t Fm(\))j(are)h(adjoin)m(t)e(to)h(the)g(orthogonal)f(p)s(olynomials)e (\010)3046 242 y Fv(n)3093 227 y Fm(\()p Fl(z)t Fm(\))k(in)f(the)g (sense)150 347 y(that)g(\010)431 311 y Fx(\003)431 372 y Fv(n)479 347 y Fm(\()p Fl(z)t Fm(\))c(=)g(1)22 b(+)g Fi(\001)17 b(\001)g(\001)j Fm(+)k(\026)-51 b Fl(c)1183 362 y Fv(j)1219 347 y Fl(z)1268 311 y Fv(n)p Fx(\000)p Fv(j)1426 347 y Fm(+)22 b Fi(\001)17 b(\001)g(\001)47 b Fm(if)31 b(\010)1848 311 y Fx(\003)1848 372 y Fv(n)1896 347 y Fm(\()p Fl(z)t Fm(\))d(=)g Fl(z)e Fm(+)d Fi(\001)17 b(\001)g(\001)j Fm(+)i Fl(c)2601 362 y Fv(j)2637 347 y Fl(z)2686 311 y Fv(j)2746 347 y Fm(+)g Fi(\001)17 b(\001)g(\001)e Fm(.)296 468 y(So)38 b(called)e(circular)g(\(re\015ection,)i(Sh)m (ur's\))h(parameters)e Fi(f)p Fl(a)2517 483 y Fv(n)2564 468 y Fi(g)2614 431 y Fx(1)2614 492 y Fv(n)p Fw(=0)2789 468 y Fm(satisfy)g Fi(j)p Fl(a)3177 483 y Fv(n)3224 468 y Fi(j)27 b Ff(6)i Fm(1)o(.)38 b(More-)150 588 y(o)m(v)m(er,)1783 756 y Fi(j)p Fl(a)1862 771 y Fv(n)1909 756 y Fi(j)27 b Fl(<)h Fm(1)1432 b(\(2.2\))150 924 y(for)32 b(all)f Fl(n)h Fm(if)g(and)g(only)g(if)g(the)h(measure)g(is)f(not)g(concen)m (trated)i(in)e(a)h(\014nite)f(n)m(um)m(b)s(er)h(of)f(atoms.)296 1045 y(Con)m(v)m(ersely)-8 b(,)32 b(if)c(conditions)g(\(2.2\))h(are)h (satis\014ed,)g(then)g(there)g(exists)g(a)f(unique)h(Borel)e(prob-)150 1165 y(abilit)m(y)40 b(measure)i Fl(\034)54 b Fm(on)42 b(the)g(unit)g(circle)f(suc)m(h)j(that)e(p)s(olynomials)d Fi(f)p Fm(\010)2873 1180 y Fv(n)2920 1165 y Fm(\()p Fl(z)t Fm(\))p Fi(g)3095 1129 y Fx(1)3095 1190 y Fv(n)p Fw(=0)3232 1165 y Fm(,)k(de\014ned)g(b)m(y)150 1286 y(\(2.1\),)32 b(are)h(orthogonal)d(with)j(resp)s(ect)g(to)g Fl(\034)11 b Fm(.)296 1406 y(The)40 b(theory)f(of)f(orthogonal)e(p)s(olynomials)g (on)i(the)h(circle)e(w)m(as)j(dev)m(elop)s(ed)f(b)m(y)g(G.)f(Szeg\177) -49 b(o,)150 1526 y(N.)30 b(I.)g(Akhiezer,)g(L.)g(Y)-8 b(a.)30 b(Geronim)m(us)e Fe(et)33 b(al)p Fm(.)c(\([A,G,Se]\).)i(The)f (follo)m(wing)d(theorem)j(is)f(a)g(com)m(bi-)150 1647 y(nation)i(of)g(results)i(of)e(G.)h(Szeg\177)-49 b(o,)32 b(A.)h(N.)f(Kolmogoro)m(v,)e(M.)i(G.)g(Krein)f(and)h(L.)g(Y)-8 b(a.)32 b(Geronim)m(us)150 1767 y(\(see)i([G]\).)150 1926 y Fg(Theorem.)49 b Fe(The)40 b(line)-5 b(ar)40 b(sp)-5 b(an)40 b(of)g Fi(f)p Fm(\010)1624 1941 y Fv(n)1672 1926 y Fm(\()p Fl(z)t Fm(\))p Fi(g)1847 1890 y Fx(1)1847 1951 y Fv(n)p Fw(=0)2025 1926 y Fe(is)g(not)h(dense)f(in)h Fl(L)2780 1890 y Fw(2)2780 1951 y Fv(\034)2864 1926 y Fe(if)f(and)g(only)h(if)g(any)f(of)150 2047 y(the)35 b(fol)5 b(lowing)33 b(\014ve)i(e)-5 b(quivalent)34 b(statements)h(hold) 228 2206 y(\(I\))1558 2303 y Fh(Z)1657 2329 y Fw(2)p Fv(\031)1613 2529 y Fw(0)1756 2439 y Fm(log)17 b Fl(\034)1952 2398 y Fx(0)1976 2439 y Fm(\()p Fl(e)2059 2398 y Fj(i)i Fv(\022)2141 2439 y Fm(\))p Fl(d\022)30 b(>)e Fi(\0001)963 b Fm(\(2.3\))394 2658 y Fe(wher)-5 b(e)32 b Fl(\034)720 2622 y Fx(0)777 2658 y Fe(is)g(the)h(density)g(of)f(the)h(absolutely)g (c)-5 b(ontinuous)32 b(c)-5 b(omp)g(onent)32 b(of)g Fl(\034)45 b Fe(with)32 b(r)-5 b(esp)g(e)g(ct)394 2778 y(to)35 b(the)g(L)-5 b(eb)g(esgue)34 b(me)-5 b(asur)g(e)35 b(on)f(the)h(unit)g(cir)-5 b(cle.)190 2967 y(\(II\))48 b(Ther)-5 b(e)34 b(exists)h(at)g(le)-5 b(ast)34 b(one)h Fl(z)k Fe(in)c(the)g(unit)g(disk)f Fl(D)d Fm(=)c Fi(f)p Fl(z)32 b Fm(:)c Fi(j)p Fl(z)t Fi(j)g Fl(<)f Fm(1)p Fi(g)35 b Fe(such)g(that)1730 3087 y Fx(1)1694 3116 y Fh(X)1699 3326 y Fv(n)p Fw(=0)1854 3211 y Fi(j)p Fm(\010)1952 3226 y Fv(n)1999 3211 y Fm(\()p Fl(z)t Fm(\))p Fi(j)2152 3170 y Fw(2)2220 3211 y Fl(<)27 b Fi(1)p Fl(:)1099 b Fm(\(2.4\))153 3501 y Fe(\(III\))47 b(Ther)-5 b(e)34 b(exists)h(at)g(le)-5 b(ast)34 b(one)h Fl(z)d Fi(2)c Fl(D)38 b Fe(such)c(that)1650 3669 y Fm(lim)17 b(inf)1693 3729 y Fv(n)p Fx(!1)1937 3669 y Fi(j)p Fm(\010)2035 3628 y Fx(\003)2035 3694 y Fv(n)2082 3669 y Fm(\()p Fl(z)t Fm(\))p Fi(j)28 b Fl(<)g Fi(1)p Fl(:)155 3904 y Fe(\(IV\))49 b(The)34 b(series)43 b Fm(\(2.4\))34 b Fe(c)-5 b(onver)g(ges)34 b(uniformly)g(on)h(c)-5 b(omp)g(act)34 b(subsets)h(of)f Fl(D)s Fe(.)193 4092 y(\(V\))49 b(Ther)-5 b(e)34 b(exists)h(a)f (function)h Fm(\005\()p Fl(z)t Fm(\))p Fe(,)g(analytic)g(in)g Fl(D)s Fe(,)f(such)h(that)g(the)g(limit)1685 4261 y Fm(\005\()p Fl(z)t Fm(\))29 b(=)51 b(lim)2015 4320 y Fv(n)p Fx(!1)2216 4261 y Fm(\010)2286 4219 y Fx(\003)2286 4285 y Fv(n)2333 4261 y Fm(\()p Fl(z)t Fm(\))1091 b(\(2.5\))394 4456 y Fe(is)34 b(uniformly)h(c)-5 b(onver)g(gent)34 b(on)g(c)-5 b(omp)g(act)35 b(subsets)f(of)h Fl(D)s Fe(.)150 4615 y(Mor)-5 b(e)g(over,)34 b(the)h(statements)g(\(I)16 b({V\))35 b(ar)-5 b(e)35 b(e)-5 b(quivalent)34 b(to)h(the)g(c)-5 b(ondition)1681 4734 y Fx(1)1644 4764 y Fh(X)1650 4973 y Fv(n)p Fw(=0)1805 4859 y Fi(j)p Fl(a)1884 4874 y Fv(n)1931 4859 y Fi(j)1959 4818 y Fw(2)2025 4859 y Fl(<)28 b Fi(1)p Fl(:)1293 b Fm(\(2.6\))296 5114 y(Note)32 b(that)g(in)f(\(I\))h(the)h (in)m(tegral)d(is)i(alw)m(a)m(ys)g(less)h(than)f(+)p Fi(1)p Fm(,)f(and)h(that)g(there)h(is)e(no)h(restric-)150 5235 y(tions)g(on)g(the)h(singular)f(part)g(of)g Fl(\034)11 b Fm(.)1926 5484 y(5)p eop %%Page: 6 6 6 5 bop 150 227 a FF(3)161 b(Krein's)54 b(theorem.)150 446 y Fm(In)43 b([K])h(M.)f(G.)g(Krein)f(studied)i(the)f(follo)m(wing)e (canonical)h(system)i(of)e(ordinary)h(di\013eren)m(tial)150 566 y(equations)1198 755 y Fl(d)p 1175 799 98 4 v 1175 890 a(dr)1315 822 y(p)17 b Fm(\()p Fl(r)m(;)g(\025)p Fm(\))22 b(=)50 b Fd(i)12 b Fl(\025)17 b(p)p Fm(\()p Fl(r)m(;)g(\025)p Fm(\))k Fi(\000)p 2256 735 174 4 v 23 w Fl(a)p Fm(\()p Fl(r)s Fm(\))16 b Fl(p)2495 781 y Fx(\003)2534 822 y Fm(\()p Fl(r)m(;)h(\025)p Fm(\))1198 993 y Fl(d)p 1175 1037 98 4 v 1175 1128 a(dr)1298 1060 y(p)1347 1019 y Fx(\003)1387 1060 y Fm(\()p Fl(r)m(;)g(\025)p Fm(\))f(=)39 b Fi(\000)22 b Fl(a)p Fm(\()p Fl(r)s Fm(\))17 b Fl(p)p Fm(\()p Fl(r)m(;)g(\025)p Fm(\))3549 932 y(\(3.1\))150 1303 y(where)36 b Fl(a)p Fm(\()p Fi(\001)p Fm(\))e(is)h(a)f(measurable) g(lo)s(cally)e(in)m(tegrable)i(function)g(on)g([0)p Fl(;)17 b Fi(1)p Fm(\),)34 b(and)h(the)g(initial)c(con-)150 1423 y(ditions)g(are)1487 1544 y Fl(p)p Fm(\(0)p Fl(;)17 b(\025)p Fm(\))27 b(=)g Fl(p)1941 1503 y Fx(\003)1981 1544 y Fm(\(0)p Fl(;)17 b(\025)p Fm(\))27 b(=)g(1)p Fl(:)150 1715 y Fm(There)i(is)f(a)g (unique)h(Borel)e(measure)i Fl(\034)39 b Fm(on)28 b Fk(R)5 b Fm(,)35 b(whic)m(h)28 b(called)f(the)i(sp)s(ectral)f(measure,)h(suc)m (h)g(that)1511 1834 y Fh(Z)1567 2059 y Fy(R)1754 1902 y Fm(1)p 1646 1947 266 4 v 1646 2038 a(1)21 b(+)h Fl(\025)1871 2009 y Fw(2)1921 1969 y Fl(d\034)11 b Fm(\()p Fl(\025)p Fm(\))27 b Fl(<)h Fi(1)1160 b Fm(\(3.2\))150 2238 y(and)33 b(the)g(map)e Fc(U)d Fm(:)g Fl(L)944 2202 y Fw(2)944 2268 y([0)p Fv(;)p Fx(1)p Fw(\))1149 2238 y Fi(!)f Fl(L)1342 2202 y Fw(2)1342 2263 y Fv(\034)1418 2238 y Fm(de\014ned)34 b(b)m(y)1385 2522 y Fc(U)p Fl(f)11 b Fm(\()p Fl(\025)p Fm(\))27 b(=)1779 2386 y Fh(Z)1878 2413 y Fx(1)1834 2612 y Fw(0)1970 2522 y Fl(f)11 b Fm(\()p Fl(r)s Fm(\))p Fl(p)p Fm(\()p Fl(r)m(;)17 b(\025)p Fm(\))p Fl(dr)1035 b Fm(\(3.3\))150 2787 y(is)31 b(an)h(isometry)-8 b(.)31 b(Con)m(v)m(ersely)-8 b(,)33 b(if)e(the)h(condition)e(\(3.2\))h(is)h(satis\014ed)g(for)f(a)g (Borel)g(measure)h Fl(\034)43 b Fm(on)150 2907 y Fk(R)5 b Fm(,)39 b(then)33 b(there)h(exists)f(a)f(unique)i(measurable)e(lo)s (cally)e(in)m(tegrable)i(function)g Fl(a)p Fm(\()p Fi(\001)p Fm(\))g(on)h Fk(R)3465 2922 y Fw(+)3562 2907 y Fm(suc)m(h)150 3027 y(that)f Fc(U)h Fm(de\014ned)h(b)m(y)h(\(3.1\))d(and)h(\(3.3\))f (is)g(an)g(isometry)g Fc(U)c Fm(:)g Fl(L)2446 2991 y Fw(2)2446 3057 y([0)p Fv(;)p Fx(1)p Fw(\))2650 3027 y Fi(!)g Fl(L)2844 2991 y Fw(2)2844 3052 y Fv(\034)2887 3027 y Fm(.)296 3157 y(A)47 b(simple)e(example)h(is)f(the)i(situation)e (when)j Fl(a)p Fm(\()p Fl(r)s Fm(\))j Fi(\021)g Fm(0)46 b(and)h Fc(U)f Fm(is)g(the)h(usual)f(F)-8 b(ourier)150 3277 y(transform.)45 b(In)h(this)f(case)i Fl(\034)57 b Fm(is)45 b(the)h(Leb)s(esgue)h(measure)f(normalized)e(b)m(y)j(2)p Fl(\031)t Fm(.)e(F)-8 b(or)45 b(a)h(more)150 3397 y(detailed)32 b(study)h(see)h([AR,R,D2,D3{5,DKu].)296 3518 y(The)d(follo)m(wing)d (imp)s(ortan)m(t)g(result)i(w)m(as)i(stated)e(b)m(y)h(M.)g(G.)f(Krein)f (in)h([K])g(without)g(a)g(pro)s(of.)150 3714 y Fg(Theorem.)49 b Fe(The)33 b(isometry)h Fc(U)g Fe(is)g(not)g(onto)f(if)h(and)f(only)h (if)g(any)g(of)f(the)h(fol)5 b(lowing)33 b(\014ve)g(e)-5 b(quiv-)150 3835 y(alent)35 b(statements)f(hold)228 4031 y(\(I\))1616 4162 y Fh(Z)1671 4388 y Fy(R)1750 4230 y Fm(log)17 b Fl(\034)1946 4194 y Fx(0)1969 4230 y Fm(\()p Fl(\025)p Fm(\))p 1750 4275 353 4 v 1793 4366 a(1)22 b(+)g Fl(\025)2019 4337 y Fw(2)2112 4298 y Fl(d\025)27 b(>)h Fi(\0001)1021 b Fm(\(3.4\))394 4561 y Fe(wher)-5 b(e)32 b Fl(\034)720 4524 y Fx(0)777 4561 y Fe(is)g(the)h(density)g(of) f(the)h(absolutely)g(c)-5 b(ontinuous)32 b(c)-5 b(omp)g(onent)32 b(of)g Fl(\034)45 b Fe(with)32 b(r)-5 b(esp)g(e)g(ct)394 4681 y(to)35 b(the)g(L)-5 b(eb)g(esgue)34 b(me)-5 b(asur)g(e)35 b(on)f Fk(R)5 b Fe(.)190 4882 y(\(II\))48 b(Ther)-5 b(e)34 b(exists)h(at)g(le)-5 b(ast)34 b(one)h Fl(\025)f Fe(in)h(the)g(domain)f Fl(H)h Fm(=)27 b Fi(f)p Fl(\025)h Fm(:)f Fe(Im)p Fl(\025)g(>)h Fm(0)p Fi(g)34 b Fe(such)h(that)1618 5009 y Fh(Z)1718 5035 y Fx(1)1673 5235 y Fw(0)1809 5145 y Fi(j)p Fl(p)p Fm(\()p Fl(r)m(;)17 b(\025)p Fm(\))p Fi(j)2132 5104 y Fw(2)2170 5145 y Fl(dr)31 b(<)c Fi(1)p Fl(:)1023 b Fm(\(3.5\))1926 5484 y(6)p eop %%Page: 7 7 7 6 bop 153 227 a Fe(\(III\))47 b(Ther)-5 b(e)34 b(exists)h(at)g(le)-5 b(ast)34 b(one)h Fl(\025)27 b Fi(2)h Fl(H)43 b Fe(such)34 b(that)1619 447 y Fm(lim)17 b(inf)1666 507 y Fv(r)r Fx(!1)1906 447 y Fi(j)p Fl(p)1983 406 y Fx(\003)2022 447 y Fm(\()p Fl(r)m(;)g(\025)p Fm(\))p Fi(j)27 b Fl(<)g Fi(1)p Fl(:)1024 b Fm(\(3.6\))155 741 y Fe(\(IV\))49 b(The)34 b(inte)-5 b(gr)g(al)45 b Fm(\(3.5\))34 b Fe(c)-5 b(onver)g(ges)34 b(uniformly)h(on)f(c)-5 b(omp)g(act)34 b(subsets)h(of)f Fl(H)8 b Fe(.)193 944 y(\(V\))49 b(Ther)-5 b(e)38 b(exists)h(a)f (function)h Fm(\005\()p Fl(\025)p Fm(\))p Fe(,)f(analytic)h(in)g Fl(H)8 b Fe(,)38 b(such)g(that)i(on)e(any)h(c)-5 b(omp)g(act)38 b(subset)394 1064 y(of)c Fl(H)43 b Fe(ther)-5 b(e)35 b(is)f(a)h(uniform)f(limit)1625 1285 y Fm(\005\()p Fl(\025)p Fm(\))28 b(=)52 b(lim)1962 1344 y Fv(n)p Fx(!1)2163 1285 y Fl(p)2212 1243 y Fx(\003)2251 1285 y Fm(\()p Fl(r)2333 1300 y Fv(n)2380 1285 y Fl(;)17 b(\025)p Fm(\))1030 b(\(3.7\))394 1531 y Fe(for)35 b(a)f(se)-5 b(quenc)g(e)34 b Fl(r)1074 1546 y Fv(n)1149 1531 y Fi(!)27 b(1)p Fe(.)296 1735 y Fm(P)m(arts)47 b(\(I)s(I)s(I,V\))f(of)g(this)f(theorem)h(w)m(ere)h(not) f(stated)h(in)e([K])h(correctly)-8 b(.)46 b(Namely)-8 b(,)45 b(it)g(w)m(as)150 1855 y(written)32 b(as)h(if)f(\(I,I)s(I,IV\))h (w)m(ere)h(equiv)-5 b(alen)m(t)32 b(to:)131 2059 y(\(I)s(I)s(I'\))49 b(There)34 b(exists)f(at)f(least)g(one)h Fl(\025)28 b Fi(2)g Fl(H)40 b Fm(suc)m(h)34 b(that)e(sup)2349 2082 y Fv(r)r Fb(>)p Fw(0)2494 2059 y Fi(j)p Fl(p)2571 2022 y Fx(\003)2610 2059 y Fm(\()p Fl(r)m(;)17 b(\025)p Fm(\))p Fi(j)27 b Fl(<)h Fi(1)o Fm(.)169 2262 y(\(V'\))49 b(The)27 b(limit)c(\005\()p Fl(\025)p Fm(\))28 b(=)f(lim)1288 2277 y Fv(r)r Fx(!1)1483 2262 y Fl(p)1532 2226 y Fx(\003)1572 2262 y Fm(\()p Fl(r)m(;)17 b(\025)p Fm(\))26 b(con)m(v)m(erges)i (uniformly)d(on)h(compact)g(subsets)i(of)e Fl(H)8 b Fm(.)150 2465 y(In)29 b(Section)g(4)f(w)m(e)i(presen)m(t)h(t)m(w)m(o)e(coun)m (terexamples.)h(The)g(corrected)g(statemen)m(t)f(app)s(eared)g(\014rst) 150 2586 y(in)j([T1,T2].)296 2706 y(In)e(Section)f(4)g(w)m(e)i(pro)m(v) m(e)g(that,)e(in)g(general,)g(\005\()p Fl(\025)p Fm(\))g(can)h(not)f(b) s(e)g(de\014ned)i(uniquely)-8 b(,)30 b(but)f(only)150 2827 y(up)40 b(to)g(a)g(factor)f(of)h(absolute)f(v)-5 b(alue)40 b(one.)g(In)g(Section)g(6)g(w)m(e)h(sho)m(w)g(that)f(if)e Fl(a)p Fm(\()p Fl(r)s Fm(\))i Fi(2)h Fl(L)3453 2790 y Fw(2)3493 2827 y Fm([0)p Fl(;)17 b Fi(1)p Fm(\))150 2947 y(then)35 b(\(I)16 b({V\))35 b(as)g(w)m(ell)f(as)h(\(I)s(I)s(I',V'\))h (hold,)e(but)h(the)g(result)g(is)f(sharp)h(in)f(the)i(sense)g(of)f (Theorem)150 3067 y(2)d(and)h(Remark)f(6.1.)296 3188 y(Note)g(that)g(in)f(\(I\))h(the)h(in)m(tegral)d(is)i(alw)m(a)m(ys)g (less)h(than)f(+)p Fi(1)p Fm(,)f(and)h(that)g(there)h(is)e(no)h (restric-)150 3308 y(tions)g(on)g(the)h(singular)f(part)g(of)g Fl(\034)11 b Fm(.)150 3641 y FF(4)161 b(Tw)l(o)53 b(results)g(on)g (noncon)l(v)l(ergence.)150 3860 y Fg(Theorem)37 b(1.)49 b Fe(Ther)-5 b(e)29 b(exists)h(a)g(r)-5 b(e)g(al-value)g(d)45 b Fl(C)1948 3824 y Fx(1)2053 3860 y Fe(function)30 b Fl(a)p Fm(\()p Fl(r)s Fm(\))f Fe(such)h(that)g(the)g(sp)-5 b(e)g(ctr)g(al)30 b(me)-5 b(a-)150 3980 y(sur)g(e)34 b Fl(\034)45 b Fe(is)34 b(absolutely)g(c)-5 b(ontinuous)33 b(with)h(p)-5 b(ositive)33 b(c)-5 b(ontinuous)34 b(density,)g (statements)f(\(I)16 b({V\))34 b(of)150 4101 y(Kr)-5 b(ein)-10 b('s)34 b(the)-5 b(or)g(em)35 b(hold,)f(but)1222 4321 y Fm(lim)17 b(inf)1269 4380 y Fv(r)r Fx(!1)1509 4321 y Fi(j)p Fl(p)1586 4279 y Fx(\003)1625 4321 y Fm(\()p Fl(r)m(;)g(\025)p Fm(\))p Fi(j)27 b Fl(<)g Fm(lim)17 b(sup)2063 4400 y Fv(r)r Fx(!1)2317 4321 y Fi(j)p Fl(p)2394 4279 y Fx(\003)2433 4321 y Fm(\()p Fl(r)m(;)g(\025)p Fm(\))p Fi(j)870 b Fm(\(4.1\))150 4592 y Fe(for)30 b(any)g Fl(\025)e Fi(2)g Fl(H)8 b Fe(.)30 b(In)f(addition,)g(the)i Fm(lim)17 b(sup)28 b Fe(in)i(\(4.1\))f(c)-5 b(an)30 b(b)-5 b(e)30 b(either)g(\014nite)g(or)g(identic)-5 b(al)5 b(ly)30 b Fm(+)p Fi(1)150 4713 y Fe(on)k Fl(H)8 b Fe(.)296 4916 y Fm(First,)32 b(w)m(e)i(need)f(the)g(follo)m(wing)d(lemma.)1926 5484 y(7)p eop %%Page: 8 8 8 7 bop 150 227 a Fg(Lemma)37 b(4.1.)49 b Fe(Supp)-5 b(ose)26 b Fl(b)p Fm(\()p Fl(r)s Fm(\))h Fe(b)-5 b(e)26 b(a)g(r)-5 b(e)g(al)27 b(nonzer)-5 b(o)25 b(c)-5 b(ontinuous)27 b(function)f(with)h(supp)-5 b(ort)26 b(in)h Fm([0)p Fl(;)17 b Fm(2])150 347 y Fe(such)39 b(that)h Fl(b)p Fm(\()p Fl(r)s Fm(\))f Fe(is)h(nonne)-5 b(gative)38 b(for)h Fl(r)f Fi(2)f Fm([0)p Fl(;)17 b Fm(1])p Fe(,)39 b(and)f Fl(b)p Fm(\()p Fl(r)s Fm(\))e(=)g Fi(\000)p Fl(b)p Fm(\()p Fl(r)29 b Fi(\000)d Fm(2\))40 b Fe(for)f Fl(r)f Fi(2)f Fm([1)p Fl(;)17 b Fm(2])p Fe(.)39 b(L)-5 b(et)150 468 y Fl(p)199 483 y Fv(")236 468 y Fm(\()p Fl(r)m(;)17 b(\025)p Fm(\))p Fe(,)34 b Fl(p)567 431 y Fx(\003)567 492 y Fv(")607 468 y Fm(\()p Fl(r)m(;)17 b(\025)p Fm(\))34 b Fe(b)-5 b(e)35 b(the)g(solutions)f(of)h(\(3.1\))f(with)1334 675 y Fl(a)p Fm(\()p Fl(r)s Fm(\))28 b(=)f Fl(a)1690 690 y Fv(")1727 675 y Fm(\()p Fl(r)s Fm(\))g(=)h Fi(\000)2068 627 y Fw(log)13 b Fx(j)f Fw(log)g Fv(")p Fx(j)p 2068 652 290 4 v 2197 709 a Fv(")2368 675 y Fl(b)p Fm(\()2457 636 y Fv(r)p 2457 652 34 4 v 2458 709 a(")2501 675 y Fm(\))p Fl(;)150 872 y Fm(0)27 b Fl(<)h(")f(<)h Fm(1)p Fe(,)34 b(and)h(initial)f(c)-5 b(onditions)34 b Fl(p)1611 887 y Fv(")1648 872 y Fm(\(0)p Fl(;)17 b(\025)p Fm(\))26 b(=)i Fl(c)p Fe(,)35 b Fl(p)2160 836 y Fx(\003)2160 896 y Fv(")2199 872 y Fm(\(0)p Fl(;)17 b(\025)p Fm(\))27 b(=)h Fl(c)2598 836 y Fx(\003)2637 872 y Fe(.)296 992 y(If)35 b Fl(\025)f Fe(is)h(any)g(c)-5 b(omplex)33 b(numb)-5 b(er,)35 b(then)587 1189 y Fl(p)636 1204 y Fv(")672 1189 y Fm(\(2)p Fl(";)17 b(\025)p Fm(\))27 b(=)g Fl(c)c Fm(+)f Fl(o)p Fm(\()1322 1113 y Fi(p)p 1404 1113 46 4 v 1404 1189 a Fl(")p Fm(\))1488 1204 y Fv(")p Fx(!)p Fw(0)1830 1189 y Fe(and)199 b Fl(p)2233 1148 y Fx(\003)2233 1214 y Fv(")2272 1189 y Fm(\(2)p Fl(";)17 b(\025)p Fm(\))27 b(=)g Fl(c)2716 1148 y Fx(\003)2778 1189 y Fm(+)22 b Fl(o)p Fm(\()2961 1113 y Fi(p)p 3043 1113 V 3043 1189 a Fl(")p Fm(\))3127 1204 y Fv(")p Fx(!)p Fw(0)3270 1189 y Fl(;)252 b Fm(\(4.2\))150 1386 y Fe(wher)-5 b(e)38 b(the)g(limits)h(ar)-5 b(e)38 b(uniform)g(for)g Fl(\025)p Fe(,)h Fl(c)p Fe(,)f Fl(c)1852 1350 y Fx(\003)1930 1386 y Fe(in)g(any)h(c)-5 b(omp)g(act)37 b(subset)i(of)f Fk(C)20 b Fe(.)45 b(If)38 b(in)g(addition)150 1507 y Fl(c)28 b Fi(6)p Fm(=)f Fi(\000)p Fl(c)442 1470 y Fx(\003)482 1507 y Fe(,)35 b(then)1237 1704 y Fm(lim)1236 1763 y Fv(")p Fx(!)p Fw(0)1391 1704 y Fi(j)p Fl(p)1468 1719 y Fv(")1504 1704 y Fm(\()p Fl(";)17 b(\025)p Fm(\))p Fi(j)27 b Fm(=)i(lim)1885 1763 y Fv(")p Fx(!)p Fw(0)2040 1704 y Fi(j)p Fl(p)2117 1662 y Fx(\003)2117 1728 y Fv(")2156 1704 y Fm(\()p Fl(";)17 b(\025)p Fm(\))p Fi(j)27 b Fm(=)g Fi(1)p Fl(:)885 b Fm(\(4.3\))150 1927 y Fe(Pr)-5 b(o)g(of.)48 b Fm(First,)32 b(w)m(e)i(consider)f(di\013eren)m(tial)d(equations)1487 2101 y Fl(d)p 1464 2145 98 4 v 1464 2237 a(dr)1587 2168 y(q)1630 2183 y Fv(")1667 2168 y Fm(\()p Fl(r)s Fm(\))23 b(=)38 b Fi(\000)23 b Fl(a)2078 2183 y Fv(")2115 2168 y Fm(\()p Fl(r)s Fm(\))16 b Fl(q)2301 2127 y Fx(\003)2297 2193 y Fv(")2340 2168 y Fm(\()p Fl(r)s Fm(\))1487 2339 y Fl(d)p 1464 2384 V 1464 2475 a(dr)1587 2407 y(q)1634 2365 y Fx(\003)1630 2431 y Fv(")1674 2407 y Fm(\()p Fl(r)s Fm(\))g(=)38 b Fi(\000)23 b Fl(a)2078 2422 y Fv(")2115 2407 y Fm(\()p Fl(r)s Fm(\))16 b Fl(q)2297 2422 y Fv(")2334 2407 y Fm(\()p Fl(r)s Fm(\))3549 2278 y(\(4.4\))150 2635 y(with)32 b(initial)d(conditions)j Fl(q)1164 2650 y Fv(")1201 2635 y Fm(\(0\))27 b(=)g Fl(c)p Fm(,)33 b Fl(q)1605 2598 y Fx(\003)1601 2659 y Fv(")1644 2635 y Fm(\(0\))28 b(=)f Fl(c)1942 2598 y Fx(\003)1982 2635 y Fm(.)32 b(Then)i(w)m(e)g(ha)m(v)m (e)1009 2882 y Fl(q)1052 2897 y Fv(")1089 2882 y Fm(\()p Fl(r)s Fm(\))22 b Fi(\000)g Fl(q)1380 2841 y Fx(\003)1376 2907 y Fv(")1420 2882 y Fm(\()p Fl(r)s Fm(\))27 b(=)16 b(\()p Fl(c)22 b Fi(\000)h Fl(c)1906 2841 y Fx(\003)1945 2882 y Fm(\))17 b(exp)2166 2772 y Fh(n)2249 2747 y(Z)2348 2773 y Fv(r)2304 2972 y Fw(0)2403 2882 y Fl(a)2454 2897 y Fv(")2491 2882 y Fm(\()p Fl(r)s Fm(\))p Fl(dr)2712 2772 y Fh(o)2777 2882 y Fl(;)1011 3143 y(q)1054 3158 y Fv(")1091 3143 y Fm(\()p Fl(r)s Fm(\))k(+)h Fl(q)1380 3102 y Fx(\003)1376 3168 y Fv(")1420 3143 y Fm(\()p Fl(r)s Fm(\))27 b(=)16 b(\()p Fl(c)22 b Fm(+)g Fl(c)1904 3102 y Fx(\003)1944 3143 y Fm(\))17 b(exp)2164 3033 y Fh(n)2252 3143 y Fi(\000)2352 3008 y Fh(Z)2452 3034 y Fv(r)2407 3233 y Fw(0)2506 3143 y Fl(a)2557 3158 y Fv(")2594 3143 y Fm(\()p Fl(r)s Fm(\))p Fl(dr)2815 3033 y Fh(o)2881 3143 y Fl(:)3549 3014 y Fm(\(4.5\))150 3393 y(Hence)34 b Fl(q)483 3408 y Fv(")520 3393 y Fm(\(2)p Fl(")p Fm(\))27 b(=)g Fl(c)33 b Fm(and)g Fl(q)1133 3357 y Fx(\003)1129 3418 y Fv(")1172 3393 y Fm(\(2)p Fl(")p Fm(\))27 b(=)g Fl(c)1515 3357 y Fx(\003)1555 3393 y Fm(.)32 b(Moreo)m(v)m(er,)369 3655 y Fl(q)412 3670 y Fv(")449 3655 y Fm(\()p Fl(")p Fm(\))27 b(=)h Fl(q)749 3614 y Fx(\003)745 3680 y Fv(")788 3655 y Fm(\()p Fl(")p Fm(\))21 b(+)h Fl(o)p Fm(\(1\))1201 3670 y Fv(")p Fx(!)p Fw(0)1371 3655 y Fm(=)1485 3588 y(1)p 1485 3632 49 4 v 1485 3723 a(2)1544 3655 y(\()p Fl(c)g Fm(+)g Fl(c)1786 3614 y Fx(\003)1825 3655 y Fm(\))17 b(exp)2045 3544 y Fh(n)2112 3655 y Fm(log)f Fi(j)h Fm(log)f Fl(")p Fi(j)22 b(\001)2586 3519 y Fh(Z)2686 3546 y Fw(1)2642 3745 y(0)2742 3655 y Fl(b)p Fm(\()p Fl(r)s Fm(\))p Fl(dr)3004 3544 y Fh(o)3092 3655 y Fm(+)g Fl(o)p Fm(\(1\))3362 3670 y Fv(")p Fx(!)p Fw(0)3504 3655 y Fl(:)150 3899 y Fm(Th)m(us)34 b(our)f(aim)d(is)i(to)h(sho)m(w)g(that)g(for)f(0)27 b Ff(6)h Fl(r)j Ff(6)d Fm(2)p Fl(")k Fm(w)m(e)h(ha)m(v)m(e)450 4096 y Fi(j)p Fl(p)527 4111 y Fv(")563 4096 y Fm(\()p Fl(r)m(;)17 b(\025)p Fm(\))22 b Fi(\000)h Fl(q)946 4111 y Fv(")983 4096 y Fm(\()p Fl(r)s Fm(\))p Fi(j)k Fm(=)g Fl(o)p Fm(\()1349 4019 y Fi(p)p 1432 4019 46 4 v 77 x Fl(")p Fm(\))1516 4111 y Fv(")p Fx(!)p Fw(0)1853 4096 y Fm(and)196 b Fi(j)p Fl(p)2283 4055 y Fx(\003)2283 4121 y Fv(")2322 4096 y Fm(\()p Fl(r)m(;)17 b(\025)p Fm(\))22 b Fi(\000)g Fl(q)2708 4055 y Fx(\003)2704 4121 y Fv(")2747 4096 y Fm(\()p Fl(r)s Fm(\))p Fi(j)27 b Fm(=)h Fl(o)p Fm(\()3114 4019 y Fi(p)p 3197 4019 V 77 x Fl(")o Fm(\))3280 4111 y Fv(")p Fx(!)p Fw(0)3423 4096 y Fl(:)296 4293 y Fm(T)-8 b(o)24 b(sho)m(w)h(this,)e(w)m(e)i(use)g(Gron)m(w)m(all's)d (lemma:)g Fe(if)k Fl(\013)q Fm(\()p Fl(r)s Fm(\))g Fe(is)h(a)f(nonne)-5 b(gative)25 b(inte)-5 b(gr)g(able)26 b(function)150 4413 y(such)32 b(that)h(for)f(some)f(c)-5 b(onstants)32 b Fl(c)1433 4428 y Fw(1)1473 4413 y Fl(;)17 b(c)1559 4428 y Fw(2)1625 4413 y Ff(>)29 b Fm(0)j Fe(we)f(have)h Fl(\013)q Fm(\()p Fl(r)s Fm(\))27 b Ff(6)h Fl(c)2535 4428 y Fw(1)2591 4333 y Fh(R)2658 4360 y Fv(r)2638 4448 y Fw(0)2712 4413 y Fl(\013)q Fm(\()p Fl(s)p Fm(\))p Fl(ds)16 b Fm(+)g Fl(c)3144 4428 y Fw(2)3183 4413 y Fe(,)33 b(then)f Fl(\013)q Fm(\()p Fl(r)s Fm(\))27 b Ff(6)150 4534 y Fl(c)192 4549 y Fw(2)231 4534 y Fl(e)276 4498 y Fv(c)307 4507 y Ft(1)342 4498 y Fv(r)380 4534 y Fe(.)296 4654 y Fm(First,)34 b(w)m(e)i(can)f (use)h(this)e(lemma)f(to)h(sho)m(w)i(that)f Fi(j)p Fl(p)2226 4669 y Fv(")2262 4654 y Fm(\()p Fl(r)m(;)17 b(\025)p Fm(\))p Fi(j)23 b Fm(+)h Fi(j)p Fl(p)2708 4618 y Fx(\003)2708 4679 y Fv(")2747 4654 y Fm(\()p Fl(r)m(;)17 b(\025)p Fm(\))p Fi(j)31 b Ff(6)h Fm(\()p Fi(j)p Fl(c)p Fi(j)22 b Fm(+)i Fi(j)p Fl(c)3461 4618 y Fx(\003)3500 4654 y Fi(j)p Fm(\))p Fl(e)3611 4618 y Fv(M)3679 4626 y Fa(")3712 4618 y Fv(r)150 4785 y Fm(where)34 b Fl(M)526 4800 y Fv(")591 4785 y Fm(=)27 b Fi(j)p Fl(\025)p Fi(j)22 b Fm(+)937 4737 y Fw(log)12 b Fx(j)g Fw(log)h Fv(")p Fx(j)p 937 4762 290 4 v 1065 4819 a Fv(")1268 4785 y Fm(max)1252 4845 y Fw(0)p Fb(6)p Fv(s)p Fb(6)p Fw(1)1482 4785 y Fl(b)p Fm(\()p Fl(s)p Fm(\).)33 b(Using)f(this)g(lemma)f(once)i(more,)f(w)m(e) h(obtain)488 5038 y Fi(j)p Fl(p)565 5053 y Fv(")602 5038 y Fm(\()p Fl(r)m(;)17 b(\025)p Fm(\))22 b Fi(\000)g Fl(q)984 5053 y Fv(")1021 5038 y Fm(\()p Fl(r)s Fm(\))p Fi(j)g Fm(+)g Fi(j)p Fl(p)1369 4997 y Fx(\003)1369 5062 y Fv(")1408 5038 y Fm(\()p Fl(r)m(;)17 b(\025)p Fm(\))k Fi(\000)i Fl(q)1794 4997 y Fx(\003)1790 5062 y Fv(")1833 5038 y Fm(\()p Fl(r)s Fm(\))p Fi(j)k Ff(6)h Fm(2)p Fl(")p Fm(\()p Fi(j)p Fl(c)p Fi(j)21 b Fm(+)h Fi(j)p Fl(c)2536 4997 y Fx(\003)2575 5038 y Fi(j)p Fm(\))p Fl(e)2686 4997 y Fw(4)p Fv(M)2789 5005 y Fa(")2823 4997 y Fv(")2887 5038 y Fm(=)28 b Fl(o)p Fm(\()3076 4961 y Fi(p)p 3158 4961 46 4 v 3158 5038 a Fl(")p Fm(\))3242 5053 y Fv(")p Fx(!)p Fw(0)3385 5038 y Fl(;)150 5235 y Fm(whic)m(h)33 b(completes)f(the)h (pro)s(of.)p 3679 5235 4 66 v 3683 5172 59 4 v 3683 5235 V 3741 5235 4 66 v 1926 5484 a(8)p eop %%Page: 9 9 9 8 bop 150 227 a Fe(Pr)-5 b(o)g(of)35 b(of)f(The)-5 b(or)g(em)34 b(1.)48 b Fm(In)30 b(this)f(pro)s(of)g Fl(n)f Fi(!)f(1)i Fm(means)h(that)f(the)h(limit)c(is)j(tak)m(en)i(o)m(v)m(er)g (p)s(ositiv)m(e)150 347 y(in)m(tegers,)i(and)g Fl(r)d Fi(!)d(1)32 b Fm(means)h(that)f(the)h(limit)d(is)i(tak)m(en)h(o)m(v)m (er)h(p)s(ositiv)m(e)e(reals.)296 468 y(W)-8 b(e)40 b(\014x)f(a)g (function)f Fl(b)p Fm(\()p Fl(r)s Fm(\))h(whic)m(h)h(satis\014es)g(the) f(conditions)f(of)h(Lemma)e(4.1.)i(Let)g Fl(a)p Fm(\()p Fl(r)s Fm(\))g(b)s(e)150 588 y(de\014ned)34 b(b)m(y)887 779 y Fl(a)p Fm(\()p Fl(r)s Fm(\))28 b(=)1229 654 y Fx(1)1192 684 y Fh(X)1198 893 y Fv(n)p Fw(=1)1336 779 y Fm(\(2)1423 738 y Fv(n)1487 779 y Fm(log)16 b Fl(n)p Fm(\))p Fl(b)p Fm(\(2)1853 738 y Fv(n)1900 779 y Fl(r)25 b Fi(\000)e Fl(n)p Fm(2)2176 738 y Fv(n)2223 779 y Fm(\))k(=)2429 654 y Fx(1)2392 684 y Fh(X)2397 893 y Fv(n)p Fw(=1)2553 779 y Fl(a)2604 794 y Fv(")2637 802 y Fa(n)2683 779 y Fm(\()p Fl(r)e Fi(\000)d Fl(n)p Fm(\))p Fl(;)150 1026 y Fm(where)40 b Fl(a)489 1041 y Fv(")526 1026 y Fm(\()p Fi(\001)p Fm(\))e(is)g(de\014ned)i(as)f(in)f(Lemma)f(4.1,)h(and)h Fl(")2146 1041 y Fv(n)2231 1026 y Fm(=)e(2)2393 990 y Fx(\000)p Fv(n)2495 1026 y Fm(.)i(This)g(sum)f(is)g(a)55 b Fl(C)3287 990 y Fx(1)3400 1026 y Fm(function)150 1146 y(since)35 b(for)e(an)m(y)i Fl(r)i Fm(the)d(sum)g(con)m(tains)g(at)g (most)g(one)g(nonzero)h(term.)e(Then)i(b)m(y)g(Lemma)e(4.1)h(w)m(e)150 1267 y(ha)m(v)m(e)1040 1387 y Fi(j)p Fl(p)1117 1346 y Fx(\003)1156 1387 y Fm(\()p Fl(n;)17 b(\025)p Fm(\))22 b Fi(\000)h Fl(p)1562 1346 y Fx(\003)1601 1387 y Fm(\()p Fl(n)f Fm(+)g(2)1866 1346 y Fw(1)p Fx(\000)p Fv(n)2003 1387 y Fl(;)17 b(\025)p Fm(\))p Fi(j)27 b Fm(=)h Fl(o)p Fm(\(2)2435 1346 y Fx(\000)p Fv(n=)p Fw(2)2607 1387 y Fm(\))2645 1402 y Fv(n)p Fx(!1)2833 1387 y Fl(:)150 1545 y Fm(Note)41 b(that)g Fl(p)663 1509 y Fx(\003)703 1545 y Fm(\()p Fl(r)m(;)17 b(\025)p Fm(\))41 b(do)s(es)g(not)h(c)m(hange)g (when)g Fl(r)i Fm(is)d(in)f(an)h(in)m(terv)-5 b(al)40 b([)p Fl(n)29 b Fm(+)f(2)3056 1509 y Fw(1)p Fx(\000)p Fv(n)3193 1545 y Fl(;)17 b(n)28 b Fm(+)f(1])42 b(since)150 1666 y Fl(a)p Fm(\()p Fl(r)s Fm(\))c(=)h(0)g(on)g(suc)m(h)h(in)m(terv) -5 b(als.)38 b(Hence)j(the)e(limit)d(lim)2203 1681 y Fv(n)p Fx(!1)2408 1666 y Fl(p)2457 1629 y Fx(\003)2496 1666 y Fm(\()p Fl(n;)17 b(\025)p Fm(\))39 b(exists)h(and)f(is)g (\014nite.)f(By)150 1786 y(the)c(same)f(argumen)m(t,)h(w)m(e)g(ha)m(v)m (e)h(lim)1535 1801 y Fv(n)p Fx(!1)1739 1786 y Fl(p)p Fm(\()p Fl(n;)17 b(\025)p Fm(\))29 b(=)h(0)j(since)2507 1747 y Fv(d)p 2490 1763 71 4 v 2490 1820 a(dr)2570 1786 y Fl(p)p Fm(\()p Fl(r)m(;)17 b(\025)p Fm(\))29 b(=)g Fd(i)12 b Fl(\025)34 b Fm(on)f(an)m(y)h(in)m(terv)-5 b(al)150 1906 y([)p Fl(n)22 b Fm(+)g(2)404 1870 y Fw(1)p Fx(\000)p Fv(n)541 1906 y Fl(;)17 b(n)22 b Fm(+)g(1],)33 b(and)g(Im)o Fl(\025)28 b(>)f Fm(0.)33 b(Note)f(that)h(lim)2084 1921 y Fv(n)p Fx(!1)2289 1906 y Fl(p)2338 1870 y Fx(\003)2377 1906 y Fm(\()p Fl(n;)17 b(\025)p Fm(\))28 b Fi(6)p Fm(=)f(0)33 b(since,)g(b)m(y)g(\(3.1\),)967 2071 y Fl(d)p 944 2115 98 4 v 944 2207 a(dr)1067 2058 y Fh(\000)1113 2138 y Fi(j)p Fl(p)1190 2097 y Fx(\003)1229 2138 y Fm(\()p Fl(r)m(;)17 b(\025)p Fm(\))p Fi(j)1475 2097 y Fw(2)1536 2138 y Fi(\000)23 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b(e)35 b(is)f(a)h(se)-5 b(quenc)g(e)34 b Fl(t)1013 4852 y Fv(n)1088 4837 y Fi(!)27 b(1)35 b Fe(such)f(that)1467 5020 y Fm(lim)1442 5080 y Fv(n)p Fx(!1)1643 5020 y Fl(p)1692 4979 y Fx(\003)1731 5020 y Fm(\()p Fl(t)1804 5035 y Fv(n)1851 5020 y Fl(;)17 b(\025)p Fm(\))28 b(=)f Fl(\022)s Fi(j)p Fm(\005\()p Fl(\025)p Fm(\))p Fi(j)p Fl(:)1091 b Fm(\(4.8\))150 5235 y Fe(In)34 b(addition,)g Fm(lim)824 5250 y Fv(r)r Fx(!1)1019 5235 y Fl(a)p Fm(\()p Fl(r)s Fm(\))28 b(=)f(0)35 b Fe(and)f Fl(a)p Fm(\()p Fl(r)s Fm(\))28 b Fi(2)g Fl(L)1959 5199 y Fv(p)1999 5235 y Fm([0)p Fl(;)17 b Fi(1)p Fm(\))34 b Fe(for)g(any)h Fl(p)28 b(>)f Fm(2)p Fe(.)1926 5484 y Fm(9)p eop %%Page: 10 10 10 9 bop 150 227 a Fe(Pr)-5 b(o)g(of.)48 b Fm(W)-8 b(e)39 b(will)c(construct)40 b(a)d(function)h Fl(a)p Fm(\()p Fl(r)s Fm(\))f(whic)m(h)i(is)e(piecewise)i(constan)m(t,)g(and)f(then)h (can)150 347 y(b)s(e)33 b(appro)m(ximated)f(b)m(y)h(a)49 b Fl(C)1211 311 y Fx(1)1318 347 y Fm(function)32 b(that)h(still)d(has)j (the)g(desired)g(prop)s(erties.)296 468 y(First,)f(note)h(that)f(the)h (system)h(of)e(di\013eren)m(tial)f(equations)1505 662 y Fl(d)p 1482 706 98 4 v 1482 797 a(dr)1606 729 y(q)t Fm(\()p Fl(r)s Fm(\))55 b(=)39 b Fi(\000)p 2045 642 174 4 v 22 w Fl(a)p Fm(\()p Fl(r)s Fm(\))17 b Fl(q)2283 688 y Fx(\003)2322 729 y Fm(\()p Fl(r)s Fm(\))1505 900 y Fl(d)p 1482 944 98 4 v 1482 1036 a(dr)1606 967 y(q)1653 926 y Fx(\003)1692 967 y Fm(\()p Fl(r)s Fm(\))f(=)39 b Fi(\000)22 b Fl(a)p Fm(\()p Fl(r)s Fm(\))17 b Fl(q)t Fm(\()p Fl(r)s Fm(\))3549 839 y(\(4.9\))150 1259 y(has)36 b(a)g(matrix)e(solution)h Fl(Q)p Fm(\()p Fl(r)s Fm(\))e(=)1447 1118 y Fh(\022)1568 1198 y Fm(cosh)17 b Fi(j)p Fl(b)p Fi(j)p Fl(r)p 2073 1103 31 3 v 2073 1159 a Fv(b)p 2053 1175 70 4 v 2053 1232 a Fx(j)p Fv(b)p Fx(j)2149 1198 y Fm(sinh)g Fi(j)p Fl(b)p Fi(j)p Fl(r)1550 1279 y Fv(b)p 1530 1295 V 1530 1352 a Fx(j)p Fv(b)p Fx(j)1626 1318 y Fm(sinh)g Fi(j)p Fl(b)p Fi(j)p Fl(r)133 b Fm(cosh)17 b Fi(j)p Fl(b)p Fi(j)p Fl(r)2483 1118 y Fh(\023)2592 1259 y Fm(if)35 b(the)h(co)s(e\016cien)m(t)h Fl(a)p Fm(\()p Fl(r)s Fm(\))c(=)g Fi(\000)p Fl(b)150 1379 y Fm(is)f(constan)m(t.)296 1500 y(No)m(w)h(let)f Fl(b)h Fm(b)s(e)g(real)f(and)h(p)s(ositiv)m(e,) 1242 1861 y Fl(a)1293 1876 y Fv(b;\030)s(;")1433 1861 y Fm(\()p Fl(r)s Fm(\))27 b(=)1687 1627 y Fh(8)1687 1717 y(>)1687 1747 y(<)1687 1926 y(>)1687 1956 y(:)1792 1709 y Fi(\000)p Fl(b;)66 b Fm(for)32 b(0)27 b Ff(6)i Fl(r)h Ff(6)e Fl(";)p 1822 1786 48 4 v 1822 1867 a(\030)t(b;)66 b Fm(for)32 b Fl(")27 b Ff(6)h Fl(r)j Ff(6)d Fm(2)p Fl(";)1862 2012 y Fm(0)p Fl(;)65 b Fm(for)32 b Fl(r)e Ff(>)e Fm(2)p Fl(";)3501 1861 y Fm(\(4.10\))150 2229 y(where)39 b Fi(j)p Fl(\030)5 b Fi(j)35 b Fm(=)i(1.)g(Then)i(the)g(solutions)d Fl(q)t Fm(\()p Fl(r)s Fm(\))g(=)h Fl(q)2010 2244 y Fv(b;\030)s(;")2150 2229 y Fm(\()p Fl(r)s Fm(\))g(and)h Fl(q)2552 2193 y Fx(\003)2591 2229 y Fm(\()p Fl(r)s Fm(\))e(=)g Fl(q)2909 2193 y Fx(\003)2905 2255 y Fv(b;\030)s(;")3045 2229 y Fm(\()p Fl(r)s Fm(\))i(of)f(\(4.9\))g(with)150 2349 y(initial)29 b(conditions)i Fl(q)t Fm(\(0\))d(=)f(0,)33 b Fl(q)1357 2313 y Fx(\003)1396 2349 y Fm(\(0\))27 b(=)h(1)k(and)h Fl(a)p Fm(\()p Fl(r)s Fm(\))27 b(=)h Fl(a)2279 2364 y Fv(b;\030)s(;")2419 2349 y Fm(\()p Fl(r)s Fm(\))k(satisfy)750 2614 y Fl(q)793 2629 y Fv(b;\030)s(;")933 2614 y Fm(\()p Fl(")p Fm(\))27 b(=)g(sinh)17 b Fl(b";)369 b(q)1902 2629 y Fv(b;\030)s(;")2042 2614 y Fm(\(2)p Fl(")p Fm(\))27 b(=)2353 2547 y(1)p 2353 2591 49 4 v 2353 2683 a(2)2412 2614 y(\(1)22 b Fi(\000)g Fl(\030)5 b Fm(\))17 b(sinh)f(2)p Fl(b";)750 2805 y(q)797 2764 y Fx(\003)793 2830 y Fv(b;\030)s(;")933 2805 y Fm(\()p Fl(")p Fm(\))27 b(=)g(cosh)18 b Fl(b";)358 b(q)1906 2764 y Fx(\003)1902 2830 y Fv(b;\030)s(;")2042 2805 y Fm(\(2)p Fl(")p Fm(\))27 b(=)g(1)22 b(+)g(\(1)g Fi(\000)p 2721 2724 48 4 v 23 w Fl(\030)t Fm(\))17 b(sinh)2997 2762 y Fw(2)3053 2805 y Fl(b":)3501 2688 y Fm(\(4.11\))150 3023 y(Let)22 b Fl(p)363 3038 y Fv(b;\030)s(;")503 3023 y Fm(\()p Fl(r)m(;)17 b(\025)p Fm(\))k(and)h Fl(p)970 2987 y Fx(\003)970 3049 y Fv(b;\030)s(;")1110 3023 y Fm(\()p Fl(r)m(;)17 b(\025)p Fm(\))k(b)s(e)h(the)g(solutions)f(of)g (\(3.1\))g(with)h(initial)c(conditions)i Fl(p)3334 3038 y Fv(b;\030)s(;")3474 3023 y Fm(\(0)p Fl(;)d(\025)p Fm(\))27 b(=)150 3143 y(0)32 b(and)h Fl(p)470 3107 y Fx(\003)470 3169 y Fv(b;\030)s(;")610 3143 y Fm(\(0)p Fl(;)17 b(\025)p Fm(\))27 b(=)g(1,)33 b(and)f Fl(a)p Fm(\()p Fl(r)s Fm(\))c(=)f Fl(a)1620 3158 y Fv(b;\030)s(;")1760 3143 y Fm(\()p Fl(r)s Fm(\).)296 3264 y(T)-8 b(o)45 b(estimate)f(these)h(solutions)f(w)m(e)i (use)f(the)g(follo)m(wing)d(form)h(of)h(Gron)m(w)m(all's)g(lemma:)e Fe(if)150 3384 y Fl(\013)q Fm(\()p Fl(r)s Fm(\))k Fe(is)h(a)g(nonne)-5 b(gative)45 b(inte)-5 b(gr)g(able)46 b(function)h(such)g(that)g(for)g (some)f Fl(c;)17 b(\014)6 b Fm(\()p Fl(r)s Fm(\))49 b Ff(>)h Fm(0)d Fe(we)g(have)150 3504 y Fl(\013)q Fm(\()p Fl(r)s Fm(\))27 b Ff(6)h Fl(c)527 3424 y Fh(R)593 3450 y Fv(r)574 3539 y Fw(0)648 3504 y Fl(\013)q 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3950 y Fx(\003)1262 4012 y Fv(b;\030)s(;")1401 3986 y Fm(\()p Fl(r)m(;)17 b(\025)p Fm(\))p Fi(j)27 b Ff(6)i Fm(1)22 b(+)g Fl(r)s Fm(\()p Fi(j)p Fl(\025)p Fi(j)f Fm(+)h Fl(b)p Fm(\))p Fl(e)2390 3950 y Fw(\()p Fx(j)p Fv(\025)p Fx(j)p Fw(+)p Fv(b)p Fw(\))p Fv(r)2676 3986 y Fl(<)28 b(const)p Fm(,)33 b(and)f(then)839 4275 y Fi(j)p Fl(p)916 4290 y Fv(b;\030)s(;")1055 4275 y Fm(\()p Fl(r)m(;)17 b(\025)p Fm(\))p Fi(j)27 b Fl(<)h(const)1693 4134 y Fh(\022)1776 4207 y Fm(1)p 1776 4252 49 4 v 1776 4343 a(2)1835 4275 y Fi(j)p Fl(\025)p Fi(j)p Fl(e)1993 4234 y Fx(j)p Fv(\025)p Fx(j)p Fv(r)2111 4275 y Fl(br)2199 4234 y Fw(2)2261 4275 y Fm(+)22 b Fl(br)2447 4134 y Fh(\023)2548 4275 y Fl(<)27 b(const)c Fi(\001)f Fl(br)m(:)150 4544 y Fm(Using)32 b(Gron)m(w)m(all's)g(lemma)e(once)j(more,)f(w)m(e)i (obtain)721 4762 y Fi(j)p Fl(p)798 4777 y Fv(b;\030)s(;")938 4762 y Fm(\()p Fl(r)m(;)17 b(\025)p Fm(\))22 b Fi(\000)g Fl(q)1320 4777 y Fv(b;\030)s(;")1460 4762 y Fm(\()p Fl(r)s Fm(\))p Fi(j)g Fm(+)g Fi(j)p Fl(p)1808 4721 y Fx(\003)1808 4786 y Fv(b;\030)s(;")1947 4762 y Fm(\()p Fl(r)m(;)17 b(\025)p Fm(\))22 b Fi(\000)g Fl(q)2333 4721 y Fx(\003)2329 4786 y Fv(b;\030)s(;")2469 4762 y Fm(\()p Fl(r)s Fm(\))p Fi(j)27 b Fl(<)h(const)22 b Fi(\001)g Fl(br)3139 4721 y Fw(2)3501 4762 y Fm(\(4.12\))150 4979 y(whic)m(h)33 b(implies)1206 5197 y Fi(j)p Fl(p)1283 5156 y Fx(\003)1283 5222 y Fv(b;\030)s(;")1423 5197 y Fm(\()p Fl(r)m(;)17 b(\025)p Fm(\))k Fi(\000)i Fl(q)1809 5156 y Fx(\003)1805 5222 y Fv(b;\030)s(;")1945 5197 y Fm(\()p Fl(r)s Fm(\))p Fi(j)k Fl(<)h(const)22 b Fi(\001)g Fl(b)2568 5156 y Fw(2)2608 5197 y Fl(r)2655 5156 y Fw(3)3501 5197 y Fm(\(4.13\))1901 5484 y(10)p eop %%Page: 11 11 11 10 bop 150 227 a Fm(b)m(y)33 b(\(3.1\).)296 347 y(W)-8 b(e)33 b(de\014ne)1349 509 y Fl(")1395 524 y Fv(n)1469 509 y Fm(=)1678 441 y(1)p 1582 486 241 4 v 1582 583 a(log)1708 540 y Fw(2)1764 583 y Fl(n)1833 509 y(;)211 b(b)2112 524 y Fv(n)2188 509 y Fm(=)2301 441 y(log)2427 398 y Fw(2)2483 441 y Fl(n)p 2301 486 V 2351 506 a Fi(p)p 2434 506 59 4 v 71 x Fl(n)150 719 y Fm(for)32 b Fl(n)c Ff(>)g Fm(3.)k(W)-8 b(e)33 b(also)f(de\014ne)i Fl(\030)1287 734 y Fv(n)1366 719 y Fm(as)f(the)g(unique)g(complex)f(n)m(um)m(b)s(er) h(suc)m(h)h(that)802 943 y Fi(j)p Fl(\030)873 958 y Fv(n)919 943 y Fi(j)28 b Fm(=)f(1)p Fl(;)212 b Fi(j)p Fm(1)21 b Fi(\000)i Fl(\030)1607 958 y Fv(n)1653 943 y Fi(j)28 b Fm(=)1918 876 y(1)p 1822 920 241 4 v 1822 1018 a(log)1948 975 y Fw(2)2004 1018 y Fl(n)2300 943 y Fm(and)228 b(Im)o Fl(\030)2844 958 y Fv(n)2919 943 y Fl(>)27 b Fm(0)p Fl(:)150 1185 y Fm(Let)33 b Fl(a)p Fm(\()p Fl(r)s Fm(\))f(b)s(e)h(de\014ned)h(b) m(y)1357 1381 y Fl(a)p Fm(\()p Fl(r)s Fm(\))28 b(=)1699 1257 y Fx(1)1662 1286 y Fh(X)1668 1496 y Fv(n)p Fw(=3)1823 1381 y Fl(a)1874 1396 y Fv(b)1904 1404 y Fa(n)1947 1396 y Fv(;\030)1998 1404 y Fa(n)2040 1396 y Fv(;")2093 1404 y Fa(n)2139 1381 y Fm(\()p Fl(r)d Fi(\000)d Fl(m)2430 1396 y Fv(n)2477 1381 y Fm(\))p Fl(;)150 1628 y Fm(where)34 b Fl(a)483 1643 y Fv(b;\030)s(;")623 1628 y Fm(\()p Fi(\001)p Fm(\))e(is)g(de\014ned)i(b)m(y)g(\(4.10\),)e(and)h Fl(m)1913 1643 y Fv(n)1993 1628 y Fm(is)f(as)h(follo)m(ws.)e(W)-8 b(e)33 b(c)m(ho)s(ose)h Fl(m)3121 1643 y Fw(2)3188 1628 y Fm(=)28 b(0)k(and)h(eac)m(h)150 1748 y Fl(m)235 1763 y Fv(n)304 1748 y Fi(\000)23 b Fl(m)489 1763 y Fv(n)p Fx(\000)p Fw(1)659 1748 y Fm(to)32 b(b)s(e)h(large)f(enough)h(so)f (that)h(for)f Fl(m)2051 1763 y Fv(n)2126 1748 y Ff(6)c Fl(r)i Ff(6)e Fl(m)2495 1763 y Fv(n)2565 1748 y Fm(+)22 b(2)p Fl(")2758 1763 y Fv(n)763 1843 y Fh(\014)763 1903 y(\014)763 1962 y(\014)763 2022 y(\014)871 1920 y Fl(p)p Fm(\()p Fl(r)m(;)17 b(\025)p Fm(\))p 806 1964 398 4 v 806 2056 a Fl(p)855 2027 y Fx(\003)895 2056 y Fm(\()p Fl(m)1018 2071 y Fv(n)1065 2056 y Fl(;)g(\025)p Fm(\))1213 1843 y Fh(\014)1213 1903 y(\014)1213 1962 y(\014)1213 2022 y(\014)1274 1987 y Fl(<)1388 1920 y(const)p 1388 1964 229 4 v 1431 1984 a Fi(p)p 1514 1984 59 4 v 72 x Fl(n)1626 1987 y(;)2060 1843 y Fh(\014)2060 1903 y(\014)2060 1962 y(\014)2060 2022 y(\014)2149 1920 y Fl(p)2198 1884 y Fx(\003)2237 1920 y Fm(\()p Fl(r)m(;)g(\025)p Fm(\))p 2103 1964 398 4 v 2103 2056 a Fl(p)2152 2027 y Fx(\003)2191 2056 y Fm(\()p Fl(m)2314 2071 y Fv(n)2362 2056 y Fl(;)g(\025)p Fm(\))2532 1987 y Fi(\000)23 b Fm(1)2681 1843 y Fh(\014)2681 1903 y(\014)2681 1962 y(\014)2681 2022 y(\014)2742 1987 y Fl(<)2855 1920 y(const)p 2855 1964 229 4 v 2940 2056 a(n)3093 1987 y(;)381 b Fm(\(4.14\))150 2224 y(and)214 2297 y Fh(\014)214 2357 y(\014)214 2416 y(\014)214 2476 y(\014)257 2374 y Fl(p)p Fm(\()p Fl(m)429 2389 y Fv(n)499 2374 y Fm(+)22 b(2)p Fl(")692 2389 y Fv(n)738 2374 y Fl(;)17 b(\025)p Fm(\))p 257 2418 620 4 v 368 2510 a Fl(p)417 2481 y Fx(\003)457 2510 y Fm(\()p Fl(m)580 2525 y Fv(n)627 2510 y Fl(;)g(\025)p Fm(\))887 2297 y Fh(\014)887 2357 y(\014)887 2416 y(\014)887 2476 y(\014)948 2441 y Fl(<)1126 2374 y(const)p 1061 2418 359 4 v 1061 2438 a Fi(p)p 1144 2438 59 4 v 72 x Fl(n)g Fm(log)f Fl(n)1430 2441 y(;)1678 2374 y(p)1727 2338 y Fx(\003)1767 2374 y Fm(\()p Fl(m)1890 2389 y Fv(n)1959 2374 y Fm(+)22 b(2)p Fl(")2152 2389 y Fv(n)2199 2374 y Fl(;)17 b(\025)p Fm(\))p 1678 2418 659 4 v 1809 2510 a Fl(p)1858 2481 y Fx(\003)1898 2510 y Fm(\()p Fl(m)2021 2525 y Fv(n)2068 2510 y Fl(;)g(\025)p Fm(\))2369 2441 y Fi(\000)23 b Fm(1)k(=)2659 2374 y Fl(const)22 b Fm(+)g Fl(o)p Fm(\(1\))3179 2389 y Fv(n)p Fx(!1)p 2659 2418 709 4 v 2875 2510 a Fl(n)17 b Fm(log)f Fl(n)3377 2441 y(;)97 b Fm(\(4.15\))150 2684 y(whic)m(h)37 b(is)f(p)s(ossible)f (b)m(y)j(\(4.11\),)d(\(4.12\))h(and)g(\(4.13\),)g(since)h Fi(j)p Fl(p)p Fm(\()p Fl(m)2592 2699 y Fv(n)2638 2684 y Fl(;)17 b(\025)p Fm(\))p Fi(j)34 b(!)f Fm(0)j(exp)s(onen)m(tially)g (as)150 2804 y(\()p Fl(m)273 2819 y Fv(n)342 2804 y Fi(\000)23 b Fl(m)527 2819 y Fv(n)p Fx(\000)p Fw(1)664 2804 y Fm(\))28 b Fi(!)f(1)p Fm(.)296 2924 y(W)-8 b(e)32 b(ha)m(v)m(e)h(that)e Fl(p)946 2888 y Fx(\003)985 2924 y Fm(\()p Fl(r)m(;)17 b(\025)p Fm(\))31 b(is)g(constan)m(t)i(for)d Fl(r)h Fi(2)d Fm([)p Fl(m)2152 2939 y Fv(n)2219 2924 y Fm(+)19 b(2)p Fl(")2409 2939 y Fv(n)2456 2924 y Fl(;)e(m)2585 2939 y Fv(n)p Fw(+1)2722 2924 y Fm(],)32 b(in)e(particular,)g Fl(p)3445 2888 y Fx(\003)3484 2924 y Fm(\()p Fl(m)3607 2939 y Fv(n)3674 2924 y Fm(+)150 3045 y(2)p Fl(")245 3060 y Fv(n)291 3045 y Fl(;)17 b(\025)p Fm(\))38 b(=)h Fl(p)632 3009 y Fx(\003)671 3045 y Fm(\()p Fl(m)794 3060 y Fv(n)p Fw(+1)932 3045 y Fl(;)17 b(\025)p Fm(\).)38 b(Hence)i(the)g(limit)35 b(lim)1982 3060 y Fv(n)p Fx(!1)2186 3045 y Fl(p)2235 3009 y Fx(\003)2275 3045 y Fm(\()p Fl(m)2398 3060 y Fv(n)2445 3045 y Fl(;)17 b(\025)p Fm(\))38 b(do)s(es)i(not)e (exists)i(b)s(ecause)g(of)150 3165 y(\(4.15\),)32 b(since)h(Im)o Fl(\030)857 3180 y Fv(n)932 3165 y Fl(>)27 b Fm(0)32 b(and)h(so)1449 3405 y(Im)1575 3338 y Fl(p)1624 3302 y Fx(\003)1664 3338 y Fm(\()p Fl(m)1787 3353 y Fv(n)1856 3338 y Fm(+)22 b(2)p Fl(")2049 3353 y Fv(n)2096 3338 y Fl(;)17 b(\025)p Fm(\))p 1575 3382 659 4 v 1706 3473 a Fl(p)1755 3445 y Fx(\003)1795 3473 y Fm(\()p Fl(m)1918 3488 y Fv(n)1965 3473 y Fl(;)g(\025)p Fm(\))2272 3405 y Fl(>)27 b Fm(0)p Fl(:)150 3640 y Fm(A)m(t)33 b(the)g(same)f(time)f (one)i(can)g(sho)m(w)h(that)1246 3713 y Fh(\014)1246 3773 y(\014)1246 3833 y(\014)1246 3893 y(\014)1289 3790 y Fl(p)1338 3754 y Fx(\003)1377 3790 y Fm(\()p Fl(m)1500 3805 y Fv(n)1569 3790 y Fm(+)22 b(2)p Fl(")1762 3805 y Fv(n)1809 3790 y Fl(;)17 b(\025)p Fm(\))p 1289 3835 V 1419 3926 a Fl(p)1468 3897 y Fx(\003)1508 3926 y Fm(\()p Fl(m)1631 3941 y Fv(n)1678 3926 y Fl(;)g(\025)p Fm(\))1957 3713 y Fh(\014)1957 3773 y(\014)1957 3833 y(\014)1957 3893 y(\014)2013 3858 y Fi(\000)22 b Fm(1)28 b Fl(<)2346 3790 y(const)p 2302 3835 316 4 v 2302 3933 a(n)17 b Fm(log)2503 3890 y Fw(2)2559 3933 y Fl(n)2627 3858 y(;)150 4110 y Fm(and)34 b(so)h(the)f(limit)d(lim)1002 4125 y Fv(r)r Fx(!1)1198 4110 y Fi(j)p Fl(p)1275 4074 y Fx(\003)1314 4110 y Fm(\()p Fl(r)m(;)17 b(\025)p Fm(\))p Fi(j)30 b Fm(=)g Fi(j)p Fm(\005\()p Fl(\025)p Fm(\))p Fi(j)j Fm(con)m(v)m(erges.) 2457 4029 y Fh(\000)2502 4110 y Fm(The)i(con)m(v)m(ergence)i(of)d(this) f(limit)150 4230 y(also)f(follo)m(ws)f(from)g(part)h(2)h(of)f(Theorem)h (3,)f(and)h(the)g(fact)f(that)g Fl(a)p Fm(\()p Fl(r)s Fm(\))h(is)f(uniformly)e(in)m(tegrable)150 4351 y(since)j Fl(a)p Fm(\()p Fl(r)s Fm(\))27 b Fi(2)h Fl(L)750 4315 y Fv(p)790 4351 y Fm([0)p Fl(;)17 b Fi(1)p Fm(\).)1075 4270 y Fh(\001)1153 4351 y Fm(Th)m(us)34 b(conditions)d(\(I)8 b({V\))33 b(of)f(Krein's)g(theorem)h(hold)e(b)m(y)j(\(1.4\).)e(It)g(is) 150 4471 y(easy)i(to)e(see)i(that)e(lim)988 4486 y Fv(r)r Fx(!1)1184 4471 y Fl(p)p Fm(\()p Fl(r)m(;)17 b(\025)p Fm(\))27 b(=)h(0.)k(Also)g(w)m(e)i(ha)m(v)m(e)g(that)1267 4656 y(lim)1243 4716 y Fv(n)p Fx(!1)1444 4656 y Fl(p)1493 4615 y Fx(\003)1532 4656 y Fm(\()p Fl(m)1655 4671 y Fv(n)p Fw(+1)1792 4656 y Fl(;)17 b(\025)p Fm(\))22 b Fi(\000)h Fl(p)2102 4615 y Fx(\003)2141 4656 y Fm(\()p Fl(m)2264 4671 y Fv(n)2311 4656 y Fl(;)17 b(\025)p Fm(\))28 b(=)f(0)p Fl(;)150 4874 y Fm(and)32 b(so)f(for)g(an)m(y)i(complex)e Fl(\022)j Fm(of)e(absolute)f(v)-5 b(alue)31 b(one)h(there)g(is)f(a)g (sequence)k Fi(f)p Fl(t)3087 4889 y Fv(n)3134 4874 y Fi(g)3184 4837 y Fx(1)3184 4898 y Fv(n)p Fw(=1)3352 4874 y Fm(suc)m(h)e(that)150 4994 y(lim)285 5009 y Fv(n)p Fx(!1)490 4994 y Fl(p)539 4958 y Fx(\003)579 4994 y Fm(\()p Fl(t)652 5009 y Fv(n)699 4994 y Fl(;)17 b(\025)p Fm(\))27 b(=)g Fl(\022)s Fi(j)p Fm(\005\()p Fl(\025)p Fm(\))p Fi(j)p Fm(.)296 5114 y(If)38 b(eac)m(h)h Fl(m)709 5129 y Fv(n)782 5114 y Fi(\000)26 b Fl(m)970 5129 y Fv(n)p Fx(\000)p Fw(1)1145 5114 y Fm(is)37 b(large)g(enough,)h(then)g Fi(k)p Fl(a)p Fm(\()p Fl(r)s Fm(\))p Fi(k)2361 5067 y Fv(p)2361 5141 y(L)2409 5122 y Fa(p)2486 5114 y Fm(=)e(2)2664 5040 y Fh(P)2768 5066 y Fx(1)2768 5143 y Fv(n)p Fw(=3)2922 5114 y Fl(n)2980 5078 y Fx(\000)p Fv(p=)p Fw(2)3162 5114 y Fm(log)3288 5071 y Fw(2)p Fv(p)p Fx(\000)p Fw(2)3470 5114 y Fl(n)p Fm(,)i(and)150 5235 y(so)33 b Fl(a)p Fm(\()p Fl(r)s Fm(\))27 b Fi(2)h Fl(L)631 5199 y Fv(p)671 5235 y Fm([0)p Fl(;)17 b Fi(1)p Fm(\))32 b(if)f(and)i(only)f(if)g Fl(p)27 b(>)h Fm(2.)p 3679 5235 4 66 v 3683 5172 59 4 v 3683 5235 V 3741 5235 4 66 v 1901 5484 a(11)p eop %%Page: 12 12 12 11 bop 150 227 a Fg(Conjecture)38 b(4.2.)49 b Fm(W)-8 b(e)28 b(conjecture)i(that)e(if)f Fl(a)p Fm(\()p Fl(r)s Fm(\))h(is)f(a)h Fe(r)-5 b(e)g(al-value)g(d)38 b Fm(function,)28 b(and)g(conditions)150 347 y(\(I)8 b({V\))32 b(of)e(Krein's)i(theorem)f (hold,)f(then)i(\005\()p Fl(\025)p Fm(\))f(is)g(unique)h(in)e(the)i (follo)m(wing)d(sense:)k(if)d Fl(t)3448 362 y Fv(n)3523 347 y Fi(!)d(1)150 468 y Fm(and)32 b(lim)474 483 y Fv(n)p Fx(!1)679 468 y Fl(p)p Fm(\()p Fl(t)801 483 y Fv(n)848 468 y Fl(;)17 b(\025)p Fm(\))27 b(=)h(0,)j(then)i(the)f(limit)c(lim) 1982 483 y Fv(n)p Fx(!1)2186 468 y Fl(p)2235 431 y Fx(\003)2275 468 y Fm(\()p Fl(t)2348 483 y Fv(n)2395 468 y Fl(;)17 b(\025)p Fm(\))27 b(=)h(\005\()p Fl(\025)p Fm(\))j(con)m(v)m(erges)j (uniformly)150 588 y(on)41 b(compact)f(subsets)j(of)e Fl(H)8 b Fm(.)40 b(If)h(true,)g(this)g(conjecture)h(implies)c(that)j (the)g(original)d(form)h(of)150 708 y(Krein's)33 b(theorem)f(holds)g (if)g Fl(a)p Fm(\()p Fl(r)s Fm(\))g(is)g(real)g(and)g(uniformly)f(in)m (tegrable)g(\(see)j(Theorem)f(3\).)150 868 y Fg(Conjecture)38 b(4.3.)49 b Fm(W)-8 b(e)37 b(conjecture)h(that)e(if)g Fl(a)p Fm(\()p Fl(r)s Fm(\))f Fi(2)g Fl(L)2278 832 y Fw(1)2278 894 y Fv(l)q(oc)2406 868 y Fm(is)h(real,)g(and)h(conditions)e (\(I)8 b({V\))37 b(of)150 989 y(Krein's)c(theorem)f(hold,)g(then)h (\005\()p Fl(\025)p Fm(\))f(is)g(the)h(limit)c(in)j(a)m(v)m(erage)i(of) e Fl(p)2693 953 y Fx(\003)2732 989 y Fm(\()p Fl(t)2805 1004 y Fv(n)2852 989 y Fl(;)17 b(\025)p Fm(\),)33 b(that)f(is,)1362 1209 y(\005\()p Fl(\025)p Fm(\))c(=)47 b(lim)1699 1269 y Fv(r)r Fx(!1)1901 1142 y Fm(1)p 1901 1186 49 4 v 1902 1277 a Fl(r)1976 1074 y Fh(Z)2076 1100 y Fv(r)2032 1299 y Fw(0)2130 1209 y Fl(p)2179 1168 y Fx(\003)2219 1209 y Fm(\()p Fl(s;)17 b(\025)p Fm(\))p Fl(ds)150 1453 y Fm(uniformly)33 b(on)i(compact)g(subsets)i(of)e Fl(H)8 b Fm(.)35 b(Here)h Fl(a)p Fm(\()p Fl(r)s Fm(\))c Fi(2)h Fl(L)2337 1417 y Fw(1)2337 1479 y Fv(l)q(oc)2463 1453 y Fm(if)h(sup)2702 1477 y Fv(r)r Fb(>)p Fw(0)2847 1373 y Fh(R)2913 1399 y Fv(r)r Fw(+1)2894 1488 y Fv(r)3058 1453 y Fi(j)p Fl(a)p Fm(\()p Fl(s)p Fm(\))p Fi(j)p Fl(ds)d(<)h Fi(1)p Fm(.)j(If)150 1574 y(true,)h(this)g(conjecture)h(also)d(implies) g(the)i(uniqueness)i(of)d(\005\()p Fl(\025)p Fm(\).)h(Note)g(that)f(in) g(the)i(situation)150 1694 y(of)c(Theorem)g(2)g(the)h(limit)c(in)i(a)m (v)m(erage)i(of)f Fl(p)1784 1658 y Fx(\003)1823 1694 y Fm(\()p Fl(t)1896 1709 y Fv(n)1944 1694 y Fl(;)17 b(\025)p Fm(\))32 b(do)s(es)i(not)f(exists)h(if)e Fl(m)2955 1709 y Fv(n)p Fw(+1)3115 1694 y Fi(\000)23 b Fl(m)3300 1709 y Fv(n)3381 1694 y Fm(are)33 b(large)150 1814 y(enough.)150 2140 y FF(5)161 b(Sakhno)l(vic)l(h's)52 b(theorem.)150 2359 y Fm(In)26 b([S1{S5])g(L.)g(A.)g(Sakhno)m(vic)m(h)h(in)m(tro)s (duced)f(and)g(studied)g(matrix)e(analogs)h(of)g(Krein's)h(system.)150 2479 y(He)33 b(considered)g(a)g(system)g(of)f(canonical)g(di\013eren)m (tial)e(equations)1127 2625 y Fl(d)p 1103 2670 98 4 v 1103 2761 a(dr)1211 2693 y(Y)21 b Fm(\()p Fl(r)m(;)c(\025)p Fm(\))27 b(=)h Fd(i)12 b Fl(\025J)d Fc(H)q Fm(\()p Fl(r)s Fm(\))p Fl(Y)20 b Fm(\()p Fl(r)m(;)d(\025)p Fm(\))p Fl(;)211 b(r)31 b Ff(>)d Fm(0)p Fl(;)150 2888 y Fm(that)k(can)h(b)s(e)g (transformed)f(b)m(y)i(a)e(c)m(hange)h(of)g(v)-5 b(ariables)31 b(to)h(a)g(system)748 3034 y Fl(d)p 725 3078 V 725 3169 a(dr)832 3101 y(P)895 3116 y Fw(1)934 3101 y Fm(\()p Fl(r)m(;)17 b(\025)p Fm(\))33 b(=)50 b Fd(i)12 b Fl(\025D)s(P)1562 3116 y Fw(1)1600 3101 y Fm(\()p Fl(r)m(;)17 b(\025)p Fm(\))22 b(+)g Fl(A)2011 3116 y Fw(1)2051 3101 y Fm(\()p Fl(r)s Fm(\))17 b Fl(P)2254 3116 y Fw(1)2292 3101 y Fm(\()p Fl(r)m(;)g(\025)p Fm(\))22 b(+)g Fl(A)2703 3060 y Fx(\003)2703 3126 y Fw(2)2743 3101 y Fm(\()p Fl(r)s Fm(\))17 b Fl(P)2946 3116 y Fw(2)2984 3101 y Fm(\()p Fl(r)m(;)g(\025)p Fm(\))748 3272 y Fl(d)p 725 3316 V 725 3407 a(dr)832 3339 y(P)895 3354 y Fw(2)934 3339 y Fm(\()p Fl(r)m(;)g(\025)p Fm(\))33 b(=)50 b Fl(A)1384 3354 y Fw(2)1423 3339 y Fm(\()p Fl(r)s Fm(\))17 b Fl(P)1626 3354 y Fw(1)1665 3339 y Fm(\()p Fl(r)m(;)g(\025)p Fm(\))3549 3211 y(\(5.1\))150 3540 y(where)35 b Fl(r)d Fi(2)e Fm([0)p Fl(;)17 b Fi(1)p Fm(\),)33 b Fl(\025)d Fi(2)g Fk(C)60 b Fm(and)33 b Fl(D)s Fm(,)h Fl(P)1610 3555 y Fw(1)1649 3540 y Fm(\()p Fl(r)m(;)17 b(\025)p Fm(\),)34 b Fl(P)1991 3555 y Fw(2)2030 3540 y Fm(\()p Fl(r)m(;)17 b(\025)p Fm(\),)34 b Fl(A)2382 3555 y Fw(1)2421 3540 y Fm(\()p Fl(r)s Fm(\),)f Fl(A)2677 3555 y Fw(2)2717 3540 y Fm(\()p Fl(r)s Fm(\))g(are)h Fl(m)23 b Fi(\002)h Fl(m)34 b Fm(matrices,)150 3660 y(and)f(the)g (initial)28 b(conditions)k(are)1423 3830 y Fl(P)1486 3845 y Fw(1)1525 3830 y Fm(\(0)p Fl(;)17 b(\025)p Fm(\))27 b(=)h Fl(P)1945 3845 y Fw(2)1984 3830 y Fm(\(0)p Fl(;)17 b(\025)p Fm(\))27 b(=)g Fl(I)2383 3845 y Fv(m)2450 3830 y Fl(;)150 3999 y Fm(the)34 b Fl(m)23 b Fi(\002)g Fl(m)33 b Fm(iden)m(tit)m(y)g(matrix.)f(It)h(is)g(assumed)h(that)f Fl(A)2233 4014 y Fw(1)2272 3999 y Fm(\()p Fl(r)s Fm(\))c(=)f Fi(\000)p Fl(A)2678 3963 y Fx(\003)2678 4024 y Fw(1)2718 3999 y Fm(\()p Fl(r)s Fm(\),)33 b(and)h Fl(D)h Fm(is)e(a)g(constan)m(t) 150 4119 y(diagonal)22 b(matrix)g(with)i(p)s(ositiv)m(e)f(diagonal)f (en)m(tries.)i(F)-8 b(unctions)24 b Fl(A)2629 4134 y Fw(1)2668 4119 y Fl(;)17 b(A)2785 4134 y Fw(2)2849 4119 y Fm(are)24 b(lo)s(cally)d(in)m(tegrable.)296 4240 y(There)34 b(is)e(a)h(Borel)e(matrix)g(v)-5 b(alued)33 b(measure)f Fl(\034)44 b Fm(on)33 b Fk(R)43 b Fm(suc)m(h)34 b(that)1511 4297 y Fh(Z)1567 4523 y Fy(R)1754 4365 y Fm(1)p 1646 4410 266 4 v 1646 4501 a(1)21 b(+)h Fl(\025)1871 4472 y Fw(2)1921 4433 y Fl(d\034)11 b Fm(\()p Fl(\025)p Fm(\))27 b Fl(<)h Fi(1)1160 b Fm(\(5.2\))150 4659 y(and)33 b(the)g(map)e Fc(U)d Fm(:)g Fl(L)944 4623 y Fw(2)944 4689 y([0)p Fv(;)p Fx(1)p Fw(\))1149 4659 y Fi(!)f Fl(L)1342 4623 y Fw(2)1342 4684 y Fv(\034)1418 4659 y Fm(de\014ned)34 b(b)m(y)1358 4901 y Fc(U)p Fl(f)11 b Fm(\()p Fl(\025)p Fm(\))28 b(=)1752 4765 y Fh(Z)1852 4791 y Fx(1)1807 4991 y Fw(0)1943 4901 y Fl(f)11 b Fm(\()p Fl(r)s Fm(\))p Fl(P)2188 4916 y Fw(1)2227 4901 y Fm(\()p Fl(r)m(;)17 b(\025)p Fm(\))p Fl(dr)1009 b Fm(\(5.3\))150 5114 y(is)32 b(an)h(isometry)-8 b(.)296 5235 y(The)34 b(follo)m(wing)c(imp)s(ortan)m(t)g(result)j(w)m(as)g(pro) m(v)m(ed)h(b)m(y)g(L.)e(A.)h(Sakhno)m(vic)m(h)h(in)e([S2{S4].)1901 5484 y(12)p eop %%Page: 13 13 13 12 bop 150 227 a Fg(Theorem.)49 b Fe(The)34 b(fol)5 b(lowing)33 b(\014ve)i(statements)f(ar)-5 b(e)35 b(e)-5 b(quivalent)228 425 y(\(I\))1540 558 y Fh(Z)1595 784 y Fy(R)1674 626 y Fm(log)16 b(det)i Fl(\034)2022 590 y Fx(0)2045 626 y Fm(\()p Fl(\025)p Fm(\))p 1674 671 505 4 v 1793 762 a(1)k(+)g Fl(\025)2019 733 y Fw(2)2188 694 y Fl(d\025)27 b(>)h Fi(\0001)945 b Fm(\(5.4\))394 958 y Fe(wher)-5 b(e)32 b Fl(\034)720 922 y Fx(0)777 958 y Fe(is)g(the)h(density)g(of)f(the)h(absolutely)g(c)-5 b(ontinuous)32 b(c)-5 b(omp)g(onent)32 b(of)g Fl(\034)45 b Fe(with)32 b(r)-5 b(esp)g(e)g(ct)394 1079 y(to)35 b(the)g(L)-5 b(eb)g(esgue)34 b(me)-5 b(asur)g(e)35 b(on)f Fk(R)5 b Fe(.)190 1281 y(\(II\))48 b(Ther)-5 b(e)34 b(exists)h(at)g(le)-5 b(ast)34 b(one)h Fl(\025)f Fe(in)h(the)g(domain)f Fl(H)h Fm(=)27 b Fi(f)p Fl(\025)h Fm(:)f Fe(Im)p Fl(\025)g(>)h Fm(0)p Fi(g)34 b Fe(such)h(that)1583 1409 y Fh(Z)1683 1436 y Fx(1)1638 1635 y Fw(0)1774 1545 y Fi(j)p Fl(P)1865 1560 y Fw(1)1904 1545 y Fm(\()p Fl(r)m(;)17 b(\025)p Fm(\))p Fi(j)2150 1504 y Fw(2)2189 1545 y Fl(dr)30 b(<)d Fi(1)p Fl(;)1005 b Fm(\(5.5\))394 1812 y Fe(wher)-5 b(e)34 b Fi(j)22 b(\001)g(j)34 b Fe(is)h(a)g(matrix)f(norm.)153 2013 y(\(III\))47 b(Ther)-5 b(e)34 b(exists)h(at)g(le)-5 b(ast)34 b(one)h Fl(\025)27 b Fi(2)h Fl(H)43 b Fe(such)34 b(that)1612 2227 y Fm(lim)17 b(inf)1659 2287 y Fv(r)r Fx(!1)1899 2227 y Fi(j)p Fl(P)1990 2242 y Fw(2)2029 2227 y Fm(\()p Fl(r)m(;)g(\025)p Fm(\))p Fi(j)27 b Fl(<)g Fi(1)p Fl(:)1017 b Fm(\(5.6\))155 2514 y Fe(\(IV\))49 b(The)34 b(inte)-5 b(gr)g(al)45 b Fm(\(5.5\))34 b Fe(c)-5 b(onver)g(ges)34 b(uniformly)h(on)f(c)-5 b(omp)g(act)34 b(subsets)h(of)f Fl(H)8 b Fe(.)193 2715 y(\(V\))49 b(Ther)-5 b(e)42 b(exists)h(a)g(matrix)g(value)-5 b(d)43 b(function)g Fm(\005\()p Fl(\025)p Fm(\))p Fe(,)f(analytic)h(in)g Fl(H)8 b Fe(,)43 b(such)g(that)g(on)g(any)394 2836 y(c)-5 b(omp)g(act)34 b(subset)h(of)f Fl(H)43 b Fe(ther)-5 b(e)35 b(is)f(a)h(uniform)f(limit)1618 3050 y Fm(\005\()p Fl(\025)p Fm(\))28 b(=)52 b(lim)1955 3109 y Fv(n)p Fx(!1)2156 3050 y Fl(P)2219 3065 y Fw(2)2258 3050 y Fm(\()p Fl(r)2340 3065 y Fv(n)2387 3050 y Fl(;)17 b(\025)p Fm(\))1023 b(\(5.7\))394 3290 y Fe(for)35 b(a)f(se)-5 b(quenc)g(e)34 b Fl(r)1074 3305 y Fv(n)1149 3290 y Fi(!)27 b(1)p Fe(.)296 3489 y Fm(P)m(arts)38 b(\(I)s(I)s(I,V\))g(of)e(this)h(theorem)g(w)m(ere)i(not) e(stated)h(in)f([S2{S4])g(correctly)g(in)g(that)g(it)f(w)m(as)150 3609 y(written)29 b(as)g(if)f(\(I,I)s(I,IV\))i(implied)c(the)k (existence)h(of)d(the)i(limit)25 b(lim)2623 3624 y Fv(r)r Fx(!1)2819 3609 y Fl(P)2882 3624 y Fw(2)2921 3609 y Fm(\()p Fl(r)m(;)17 b(\025)p Fm(\).)29 b(Since)g(Krein's)150 3729 y(system)k(is)e(a)h(particular)e(case)j(of)e(Sakhno)m(vic)m(h's)j (system,)f(the)f(coun)m(terexamples)h(of)e(Section)h(4)150 3850 y(applies)25 b(to)h(this)g(situation)e(also.)i(It)g(is)g(easy)h (to)f(construct)h(a)f(\\true")g(matrix-v)-5 b(alued)24 b(coun)m(terex-)150 3970 y(ample)31 b(along)g(the)i(lines)f(of)g (Section)h(4.)296 4090 y(In)h(Section)g(4)g(w)m(e)g(pro)m(v)m(e)h (that,)f(in)f(general,)g(\005\()p Fl(\025)p Fm(\))h(can)g(not)g(b)s(e)g (de\014ned)h(uniquely)-8 b(.)33 b(In)i(Sec-)150 4211 y(tion)c(6)g(w)m(e)i(sho)m(w)g(that)f(if)e Fl(A)1186 4226 y Fw(2)1226 4211 y Fm(\()p Fl(r)s Fm(\))d Fi(2)h Fl(L)1536 4175 y Fw(2)1576 4211 y Fm([0)p Fl(;)17 b Fi(1)p Fm(\),)31 b(then)i(the)f(\014nite)f(limit)e(\005\()p Fl(\025)p Fm(\))e(=)h(lim)3234 4226 y Fv(r)r Fx(!1)3430 4211 y Fl(P)3493 4226 y Fw(2)3532 4211 y Fm(\()p Fl(r)m(;)17 b(\025)p Fm(\))150 4331 y(exists,)33 b(and)g(so)g(\005\()p Fl(\025)p Fm(\))f(is)g(unique.)296 4452 y(Belo)m(w)39 b(w)m(e)g(giv)m(e)f(a)g(corrected)i(part)e(of)g(the)g(pro)s(of)g(of)g (Sakhno)m(vic)m(h's)i(theorem.)e(F)-8 b(ollo)m(wing)150 4572 y(the)35 b(lines)e(of)h([S2{S4],)g(will)e(sho)m(w)k(that)e (statemen)m(ts)h(\(I)s(I)8 b({V\))35 b(are)f(equiv)-5 b(alen)m(t.)34 b(An)h(alternativ)m(e)150 4692 y(approac)m(h)e(can)g(b)s (e)g(found)f(in)g([T2].)150 4886 y Fe(A)j(pr)-5 b(o)g(of)35 b(of)f(a)h(p)-5 b(art)35 b(of)f(Sakhnovich's)f(the)-5 b(or)g(em.)48 b Fm(One)33 b(can)g(see)h(that)430 5145 y Fl(P)507 5104 y Fx(\003)493 5169 y Fw(2)545 5145 y Fm(\()p Fl(r)m(;)17 b(\025)p Fm(\))p Fl(P)826 5160 y Fw(2)866 5145 y Fm(\()p Fl(r)m(;)g(\025)p Fm(\))k Fi(\000)i Fl(P)1282 5104 y Fx(\003)1268 5169 y Fw(1)1321 5145 y Fm(\()p Fl(r)m(;)17 b(\025)p Fm(\))p Fl(P)1602 5160 y Fw(1)1641 5145 y Fm(\()p Fl(r)m(;)g(\025)p Fm(\))27 b(=)h(2Im)o Fl(\025)2229 5009 y Fh(Z)2328 5035 y Fv(r)2284 5235 y Fw(0)2383 5145 y Fl(P)2460 5104 y Fx(\003)2446 5169 y Fw(1)2499 5145 y Fm(\()p Fl(s;)17 b(\025)p Fm(\))p Fl(D)s(P)2869 5160 y Fw(1)2907 5145 y Fm(\()p Fl(s;)g(\025)p Fm(\))p Fl(ds;)295 b Fm(\(5.8\))1901 5484 y(13)p eop %%Page: 14 14 14 13 bop 150 227 a Fm(and)33 b(so)f(\(I)s(I\))h(and)g(\(I)s(I)s(I\))f (are)h(equiv)-5 b(alen)m(t.)296 347 y(Clearly)d(,)27 b(b)m(y)h(\(5.8\),)e(\(IV{V\))i(implies)c(\(I)s(I)8 b({I)s(I)s(I\).)27 b(So)h(w)m(e)g(ha)m(v)m(e)g(to)f(sho)m(w)h(that)f(\(I)s(I)8 b({I)s(I)s(I\))27 b(implies)150 468 y(\(IV{V\).)k(No)m(w)h(assume)f (that)g(\(I)s(I)8 b({I)s(I)s(I\))30 b(hold)h(for)f(some)h Fl(\025)c Fm(=)h Fl(\025)2478 483 y Fw(0)2545 468 y Fi(2)g Fl(H)8 b Fm(.)30 b(By)i(\(5.1\))e(and)h(\(5.8\),)g(the)150 588 y(family)i Fi(fj)p Fl(P)592 603 y Fw(2)631 588 y Fm(\()p Fl(r)m(;)17 b(\025)p Fm(\))p Fi(j)32 b Fm(:)h Fl(r)i Ff(>)e Fm(0)p Fl(;)17 b(\025)32 b Fi(2)i Fl(S)6 b Fi(g)35 b Fm(is)g(uniformly)e(b)s(ounded)j(from)f(b)s(elo)m(w)g(for)g (an)m(y)h(compact)150 708 y Fl(S)42 b Fi(\032)37 b Fl(H)8 b Fm(.)37 b(By)h(\(5.6\))f(and)g(Mon)m(tel's)i(theorem,)e(there)h (exists)h(a)e(sequence)j Fl(r)2990 723 y Fv(n)3073 708 y Fi(!)c(1)h Fm(suc)m(h)i(that)150 829 y(there)30 b(is)g(a)f(uniformly) f(con)m(v)m(ergen)m(t)k(limit)26 b(\(5.7\).)j(Th)m(us)j(\(V\))d(holds,) h(and)g(so)g(do)s(es)g(\(IV\))g(b)s(ecause)150 949 y(of)i(\(5.8\).)p 3679 949 4 66 v 3683 887 59 4 v 3683 949 V 3741 949 4 66 v 150 1140 a(Note)h(that)282 1372 y Fl(P)359 1331 y Fx(\003)345 1397 y Fw(2)398 1372 y Fm(\()p Fl(r)m(;)17 b(\025)578 1387 y Fw(0)617 1372 y Fm(\))p Fl(P)718 1387 y Fw(2)757 1372 y Fm(\()p Fl(r)m(;)g(\025)p Fm(\))22 b Fi(\000)h Fl(P)1174 1331 y Fx(\003)1160 1397 y Fw(1)1212 1372 y Fm(\()p Fl(r)m(;)17 b(\025)1392 1387 y Fw(0)1432 1372 y Fm(\))p Fl(P)1533 1387 y Fw(1)1572 1372 y Fm(\()p Fl(r)m(;)g(\025)p Fm(\))27 b(=)h(2\()p 2008 1291 97 4 v Fl(\025)2065 1387 y Fw(0)2126 1372 y Fi(\000)23 b Fl(\025)p Fm(\))2338 1237 y Fh(Z)2437 1263 y Fv(r)2392 1462 y Fw(0)2491 1372 y Fl(P)2568 1331 y Fx(\003)2554 1397 y Fw(1)2607 1372 y Fm(\()p Fl(s;)17 b(\025)2792 1387 y Fw(0)2831 1372 y Fm(\))p Fl(D)s(P)3016 1387 y Fw(1)3055 1372 y Fm(\()p Fl(s;)g(\025)p Fm(\))p Fl(ds;)147 b Fm(\(5.9\))150 1631 y(and)33 b(so,)g(if)e(\(I)8 b({V\))33 b(hold,)f(then)820 1888 y(\005\()p Fl(\025)p Fm(\))27 b(=)h(2\()p 1244 1807 V Fl(\025)1301 1903 y Fw(0)1362 1888 y Fi(\000)23 b Fl(\025)p Fm(\))1557 1808 y Fh(\000)1602 1888 y Fm(\005)1675 1847 y Fx(\003)1715 1888 y Fm(\()p Fl(\025)1810 1903 y Fw(0)1849 1888 y Fm(\))1887 1808 y Fh(\001)1932 1830 y Fx(\000)p Fw(1)2043 1753 y Fh(Z)2143 1779 y Fv(r)2099 1978 y Fw(0)2198 1888 y Fl(P)2275 1847 y Fx(\003)2261 1913 y Fw(1)2313 1888 y Fm(\()p Fl(s;)17 b(\025)2498 1903 y Fw(0)2537 1888 y Fm(\))p Fl(D)s(P)2722 1903 y Fw(1)2761 1888 y Fm(\()p Fl(s;)g(\025)p Fm(\))p Fl(ds)420 b Fm(\(5.10\))150 2147 y(pro)m(vided)33 b(lim)687 2162 y Fv(n)p Fx(!1)891 2147 y Fl(P)968 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y(theorem)32 b(do)h(not)f(hold.)296 1262 y(Also)41 b(this)h(result)g(is)f(sharp)h(in)f(a)h(more)f(delicate) f(sense:)k(b)m(y)f(Theorem)f(2)f(there)i(exists)f(a)150 1382 y(co)s(e\016cien)m(t)37 b Fl(A)682 1397 y Fw(2)721 1382 y Fm(\()p Fl(r)s Fm(\),)f(whic)m(h)g(is)g(again)f(in)g Fl(L)1738 1346 y Fv(p)1814 1382 y Fm(for)g(an)m(y)i Fl(p)d(>)f Fm(2,)j(suc)m(h)i(that)d(lim)3032 1397 y Fv(r)r Fx(!1)3211 1382 y Fl(P)3274 1397 y Fw(1)3313 1382 y Fm(\()p Fl(r)m(;)17 b(\025)p Fm(\))34 b(=)f(0,)150 1503 y(statemen)m(ts)42 b(\(I)8 b({V\))41 b(of)g(Sakhno)m(vic)m(h's)i(theorem)e(hold,)f(but)i (the)f(limit)d(lim)3022 1518 y Fv(r)r Fx(!1)3201 1503 y Fl(P)3264 1518 y Fw(2)3304 1503 y Fm(\()p Fl(r)m(;)17 b(\025)p Fm(\))40 b(do)s(es)150 1623 y(not)c(exist.)g(In)g(fact,)g(w)m (e)g(sho)m(w)h(that)f(\005\()p Fl(\025)p Fm(\))g(can)g(not)f(b)s(e)h (de\014ned)i(uniquely)-8 b(,)35 b(but)h(only)g(up)g(to)f(a)150 1744 y(constan)m(t)k(factor)f(ev)m(en)i(though)e(the)h(limit)c(lim)1946 1759 y Fv(r)r Fx(!1)2125 1744 y Fi(j)p Fl(P)2216 1759 y Fw(2)2255 1744 y Fm(\()p Fl(r)m(;)17 b(\025)p Fm(\))p Fi(j)37 b Fm(=)g Fi(j)p Fm(\005\()p Fl(\025)p Fm(\))p Fi(j)h Fm(exists)h(b)m(y)g(part)f(2)g(of)150 1864 y(Theorem)33 b(3.)296 1984 y(Note)39 b(that)f(in)f(statemen)m(ts)i(2)f(and)g(3)g (there)h(is)f(no)g(restriction)f(on)h(the)h(co)s(e\016cien)m(t)g Fl(A)3561 1999 y Fw(1)3600 1984 y Fm(\()p Fl(r)s Fm(\),)150 2105 y(except)34 b(for)e(the)h(usual)f(assumption)g(of)h(lo)s(cal)d(in) m(tegrabilit)m(y)-8 b(.)296 2273 y(It)53 b(w)m(as)h(pro)m(v)m(ed)g(in)d ([S2,S3])i(that)g(if)e Fl(A)1854 2288 y Fw(2)1894 2273 y Fm(\()p Fl(r)s Fm(\))61 b Fi(2)h Fl(L)2272 2237 y Fw(1)2312 2273 y Fm([0)p Fl(;)17 b Fi(1)p Fm(\),)52 b(then)h(conditions)f(\(I)8 b({V\))52 b(of)150 2394 y(Sakhno)m(vic)m(h's)33 b(theorem)e(hold,)g (and)g(the)h(limits)c(\(6.2\))j(con)m(v)m(erge)i(uniformly)c(on)j (compact)f(sub-)150 2514 y(sets)47 b(of)e Fl(H)38 b Fi([)31 b Fk(R)57 b Fm(and)45 b Fl(H)53 b Fm(resp)s(ectiv)m(ely)-8 b(.)46 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(tegral)e(\(5.5\))g(con)m(v)m(erges)k(for)d(an)m(y)h Fl(\025)29 b Fi(2)g Fl(H)8 b Fm(.)150 347 y(Let)33 b(us)g(assume)g (that)998 267 y Fh(R)1065 293 y Fx(1)1045 382 y Fw(0)1156 347 y Fi(j)p Fl(P)1247 362 y Fw(1)1286 347 y Fm(\()p Fl(r)m(;)17 b(\025)p Fm(\))p Fi(j)1532 311 y Fw(2)1571 347 y Fl(dr)30 b Fm(=)e Fi(1)k Fm(for)g(some)g Fl(\025)c Fi(2)g Fl(H)8 b Fm(.)32 b(Then)i(one)e(can)h(see)h(that)525 628 y Fi(j)p Fl(P)616 643 y Fw(2)655 628 y Fm(\()p Fl(r)m(;)17 b(\025)p Fm(\))p Fi(j)901 586 y Fw(2)967 628 y Ff(6)1073 517 y Fh(\020)1149 492 y(Z)1248 518 y Fv(r)1204 718 y Fw(0)1303 628 y Fi(j)p Fl(A)1404 643 y Fw(2)1443 628 y Fm(\()p Fl(s)p Fm(\))p Fi(jj)p Fl(P)1684 643 y Fw(1)1723 628 y Fm(\()p Fl(s;)g(\025)p Fm(\))p Fi(j)p Fl(ds)2088 517 y Fh(\021)2145 539 y Fw(2)2213 628 y Fm(=)27 b Fl(o)2363 517 y Fh(\020)2439 492 y(Z)2539 518 y Fv(r)2494 718 y Fw(0)2593 628 y Fi(j)p Fl(P)2684 643 y Fw(1)2723 628 y Fm(\()p Fl(s;)17 b(\025)p Fm(\))p Fi(j)2974 586 y Fw(2)3013 628 y Fl(ds)3110 517 y Fh(\021)3169 697 y Fv(r)r Fx(!1)3348 628 y Fl(;)150 900 y Fm(whic)m(h)33 b(con)m(tradicts)g(to)f(\(5.8\).)g (Th)m(us)i(the)f(in)m(tegral)e(\(5.5\))h(con)m(v)m(erges)j(for)d(an)m (y)i Fl(\025)27 b Fi(2)h Fl(H)40 b Fm(and)33 b(so)923 1171 y(\005\()p Fl(\025)p Fm(\))28 b(=)47 b(lim)1261 1231 y Fv(r)r Fx(!1)1452 1171 y Fl(P)1515 1186 y Fw(2)1554 1171 y Fm(\()p Fl(r)m(;)17 b(\025)p Fm(\))28 b(=)f Fl(I)1946 1186 y Fv(m)2035 1171 y Fm(+)2133 1035 y Fh(Z)2233 1062 y Fx(1)2188 1261 y Fw(0)2324 1171 y Fl(A)2397 1186 y Fw(2)2437 1171 y Fm(\()p Fl(r)s Fm(\))p Fl(P)2623 1186 y Fw(1)2661 1171 y Fm(\()p Fl(r)m(;)17 b(\025)p Fm(\))p Fl(dr)150 1448 y Fm(holds,)32 b(since)h Fl(A)744 1463 y Fw(2)784 1448 y Fm(\()p Fl(r)s Fm(\))27 b Fi(2)h Fl(L)1094 1412 y Fw(2)1134 1448 y Fm([0)p Fl(;)17 b Fi(1)p Fm(\).)32 b(The)h(rest)h(of)e(the)h(pro)s(of)e(follo)m(ws)h(from)f(\(5.8\),)h (\(5.9\).)p 3679 1448 4 66 v 3683 1385 59 4 v 3683 1448 V 3741 1448 4 66 v 150 1781 a FF(References)150 2000 y Fm([A])122 b(N.)44 b(I.)f(Akhiezer,)i Fe(The)f(classic)-5 b(al)43 b(moment)h(pr)-5 b(oblem)44 b(and)g(some)g(r)-5 b(elate)g(d)45 b(questions)f(in)399 2120 y(analysis.)k Fm(Kemmer)32 b(Hafner)g(Publishing)g(Co.,)h(New)g(Y)-8 b(ork)33 b(1965.)150 2323 y([AR])50 b(N.)33 b(I.)f(Akhiezer)i(and)e(A.) h(M.)f(Rybalk)m(o,)h Fe(Continual)h(analo)-5 b(gues)34 b(of)g(p)-5 b(olynomials)33 b(ortho)-5 b(g-)399 2444 y(onal)34 b(on)h(a)g(cir)-5 b(cle)55 b Fm(\(Russian\),)33 b(Ukrain.)f(Mat.)h(Z.)f Fg(20)g Fm(\(1968\),)g(3{24.)150 2647 y([D1])71 b(S.)25 b(A.)f(Deniso)m(v,)h Fe(On)i(the)g(applic)-5 b(ation)26 b(of)h(some)g(of)g(M.)h(G.)f(Kr)-5 b(ein)-10 b('s)27 b(r)-5 b(esults)28 b(to)f(the)g(sp)-5 b(e)g(ctr)g(al)399 2768 y(analysis)38 b(of)h(Sturm-Liouvil)5 b(le)39 b(op)-5 b(er)g(ators,)52 b Fm(J.)37 b(Math.)h(Anal.)e(Appl.)h Fg(261)g Fm(\(2001\),)f(177{)399 2888 y(191.)150 3091 y([D2])71 b(S.)35 b(A.)f(Deniso)m(v,)h Fe(T)-7 b(o)35 b(the)i(sp)-5 b(e)g(ctr)g(al)36 b(the)-5 b(ory)37 b(of)f(Kr)-5 b(ein)36 b(system,)50 b Fm(In)m(tegral)34 b(Equations)g(Op-)399 3212 y(erator)e(Theory)k Fg(42)c Fm(\(2002\),)g(166{173.)150 3415 y([D3])71 b(S.)25 b(A.)f(Deniso)m(v,)h Fe(On)i(the)g(c)-5 b(ontinuous)27 b(analo)-5 b(g)26 b(of)h(R)-5 b(akhmanov's)26 b(the)-5 b(or)g(em)27 b(for)g(ortho)-5 b(gonal)399 3536 y(p)g(olynomials,)48 b Fm(preprin)m(t.)150 3739 y([D4])71 b(S.)32 b(A.)g(Deniso)m(v,)g Fe(On)i(the)g(existenc)-5 b(e)33 b(of)h(the)g(absolutely)g(c)-5 b(ontinuous)34 b(c)-5 b(omp)g(onent)33 b(for)h(the)399 3859 y(me)-5 b(asur)g(e)35 b(asso)-5 b(ciate)g(d)33 b(with)i(some)f(ortho)-5 b(gonal)34 b(system,)49 b Fm(preprin)m(t.)150 4063 y([D5])71 b(S.)42 b(A.)g(Deniso)m(v,)g Fe(On)h(the)g(c)-5 b(o)g(existenc)g(e)43 b(of)g(absolutely)g(c)-5 b(ontinuous)43 b(and)g(singular)g(c)-5 b(on-)399 4183 y(tinuous)34 b(c)-5 b(omp)g(onents)32 b(of)h(the)h(sp)-5 b(e)g(ctr)g(al)33 b(me)-5 b(asur)g(e)33 b(for)g(some)f(Sturm-Liouvil)5 b(le)33 b(op)-5 b(er)g(ators)399 4303 y(with)35 b(squar)-5 b(e)35 b(summable)e(p)-5 b(otential,)49 b Fm(preprin)m(t.)150 4507 y([DKu])g(S.)35 b(A.)h(Deniso)m(v)g(and)f (S.)h(Kupin,)73 b Fe(On)37 b(singular)g(sp)-5 b(e)g(ctrum)38 b(of)f(Schr)-5 b(o)g(dinger)36 b(op)-5 b(er)g(ators)399 4627 y(with)35 b(de)-5 b(c)g(aying)34 b(p)-5 b(otential,)48 b Fm(preprin)m(t.)150 4831 y([G])118 b(L.)49 b(Y)-8 b(a.)49 b(Geronim)m(us,)f Fe(Ortho)-5 b(gonal)48 b(p)-5 b(olynomials:)49 b(Estimates,)g(asymptotic)h(formulas,)399 4951 y(and)f(series)g(of)h(p) -5 b(olynomials)48 b(ortho)-5 b(gonal)49 b(on)g(the)h(unit)g(cir)-5 b(cle)49 b(and)g(on)g(an)h(interval.)399 5071 y Fm(Authorized)33 b(translation)e(from)g(Russian.)i(Consultan)m(ts)g(Bureau,)g(New)g(Y)-8 b(ork)33 b(1961.)1901 5484 y(16)p eop %%Page: 17 17 17 16 bop 150 227 a Fm([K])119 b(M.)38 b(G.)e(Krein,)h Fe(Continuous)h(analo)-5 b(gues)38 b(of)h(pr)-5 b(op)g(ositions)38 b(on)g(p)-5 b(olynomials)38 b(ortho)-5 b(gonal)399 347 y(on)34 b(the)g(unit)h(cir)-5 b(cle)55 b Fm(\(Russian\),)48 b(Dokl.)31 b(Ak)-5 b(ad.)32 b(Nauk)g(SSSR)h(\(N.S.\))f Fg(105)g Fm(\(1955\),)f(637{)399 468 y(640.)150 671 y([R])123 b(A.)29 b(M.)h(Rybalk)m(o,)f Fe(On)i(the)g(the)-5 b(ory)32 b(of)g(c)-5 b(ontinual)31 b(analo)-5 b(gues)30 b(of)h(ortho)-5 b(gonal)31 b(p)-5 b(olynomials)399 791 y Fm(\(Russian\),)33 b(T)-8 b(eor.)33 b(F)-8 b(unk)m(cii)31 b(F)-8 b(unk)m(cional.)32 b(Anal.)g(i)g(Prilozen.)f(Vyp.)j(3)e(\(1966\))g(42{60.)150 995 y([S1])92 b(L.)35 b(A.)g(Sakhno)m(vic)m(h,)h Fe(Sp)-5 b(e)g(ctr)g(al)36 b(analysis)g(of)h(a)f(class)g(of)h(c)-5 b(anonic)g(al)35 b(di\013er)-5 b(ential)36 b(systems)399 1115 y Fm(\(Russian\),)i(Algebra)g(i)f(Analiz)f Fg(10)j Fm(\(1998\),)e(187{201;)f(translation)h(in)g(St.)h(P)m(etersburg)399 1236 y(Math.)33 b(J.)g Fg(10)f Fm(\(1999\),)g(147{158.)150 1439 y([S2])92 b(L.)31 b(A.)g(Sakhno)m(vic)m(h,)i Fe(On)g(a)g(class)g (of)g(c)-5 b(anonic)g(al)32 b(systems)h(on)g(half-axis,)46 b Fm(In)m(tegral)31 b(Equa-)399 1559 y(tions)h(Op)s(erator)g(Theory)j Fg(31)e Fm(\(1998\),)e(no.)i(1,)f(92{112.)150 1763 y([S3])92 b(L.)52 b(A.)g(Sakhno)m(vic)m(h,)i Fe(Sp)-5 b(e)g(ctr)g(al)52 b(the)-5 b(ory)53 b(of)g(c)-5 b(anonic)g(al)51 b(di\013er)-5 b(ential)52 b(systems.)g(Metho)-5 b(d)399 1883 y(of)59 b(op)-5 b(er)g(ator)59 b(identities.)75 b Fm(Op)s(erator)59 b(Theory:)h(Adv)-5 b(ances)61 b(and)e(Applications,)f(107.)399 2003 y(Birkh\177)-49 b(auser)33 b(V)-8 b(erlag,)32 b(Basel,)g(1999.)150 2207 y([S4])92 b(L.)46 b(A.)h(Sakhno)m(vic)m(h,)g Fe(On)g(the)h(sp)-5 b(e)g(ctr)g(al)46 b(the)-5 b(ory)48 b(of)f(a)g(class)g(of)g(c)-5 b(anonic)g(al)46 b(di\013er)-5 b(ential)399 2327 y(systems)54 b Fm(\(Russian\),)30 b(F)-8 b(unktsional.)29 b(Anal.)g(i)g(Prilozhen.)h Fg(34)g Fm(\(2000\),)f(50{62,)g(96;)g(trans-)399 2448 y(lation)i(in)g(F)-8 b(unct.)33 b(Anal.)f(Appl.)g Fg(34)h Fm(\(2000\),)e(119{128.)150 2651 y([S5])92 b(L.)37 b(A.)f(Sakhno)m(vic) m(h,)i Fe(On)g(the)g(sp)-5 b(e)g(ctr)g(al)38 b(the)-5 b(ory)39 b(of)f(the)h(gener)-5 b(alize)g(d)37 b(di\013er)-5 b(ential)37 b(system)399 2771 y(of)32 b(M.)h(G.)f(Kr)-5 b(ein)53 b Fm(\(Russian\),)30 b(Ukrain.)f(Mat.)h(Zh.)f Fg(52)h Fm(\(2000\),)f(717{721;)f(translation)g(in)399 2892 y(Ukrainian)j(Math.)i(J.)g Fg(52)g Fm(\(2000\),)e(821{826)g (\(2001\).)150 3095 y([Se])98 b(G.)29 b(Szeg\177)-49 b(o,)29 b Fe(Ortho)-5 b(gonal)30 b(p)-5 b(olynomials.)43 b Fm(F)-8 b(ourth)29 b(edition.)e(American)h(Mathematical)f(So-)399 3216 y(ciet)m(y)-8 b(,)45 b(Collo)s(quium)c(Publications,)i(V)-8 b(ol.)43 b(XXI)s(I)s(I.)h(American)g(Mathematical)e(So)s(ciet)m(y)-8 b(,)399 3336 y(Pro)m(vidence,)34 b(R.I.,)f(1975.)150 3539 y([T1])76 b(A.)34 b(T)-8 b(eply)m(aev,)35 b Fe(Continuous)h(analo) -5 b(gues)35 b(of)h(r)-5 b(andom)35 b(p)-5 b(olynomials)35 b(that)h(ar)-5 b(e)36 b(ortho)-5 b(gonal)399 3660 y(on)45 b(the)g(cir)-5 b(cle)67 b Fm(\(Russian\),)44 b(T)-8 b(eor.)44 b(V)-8 b(ero)m(y)m(atnost.)45 b(i)d(Primenen.)i Fg(39)g Fm(\(1994\),)e(588{604;)399 3780 y(translation)31 b(in)h(Theory)h (Probab.)g(Appl.)g Fg(39)f Fm(\(1994\),)g(476{489)f(\(1995\).)150 3984 y([T2])76 b(A.)30 b(T)-8 b(eply)m(aev,)30 b Fe(R)-5 b(andom)31 b(Ortho)-5 b(gonal)31 b(on)g(the)h(Cir)-5 b(cle)31 b(Polynomials)f(and)i(Their)f(Contin-)399 4104 y(uous)37 b(A)n(nalo)-5 b(gues)58 b Fm(\(Russian\),)34 b(Diploma)d(w)m(ork)k(at)f(Leningrad)g(\(St.-P)m(etersburg\))i(State) 399 4224 y(Univ)m(ersit)m(y)-8 b(,)33 b(1990.)1901 5484 y(17)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0203121354259--